THE PHENOMENON OF CATALYSIS

The phenomenon of catalysis, a tremendous (by a thousand to a million times) increase in reaction rate, is one of the most important properties of enzymatic reactions [1, 2]. From the thermodynamic point of view, catalysis is the result of the action of two factors: enthalpy (the energy factor) and entropy (the steric factor) [3, 4]. The first of these consists in the energy gain from neutralization (partial or complete) of the opposite charges as well as from the formation of ion and hydrogen bonds when the substrate binds to the enzyme. The second factor is that in binding to the enzyme the substrate becomes to a large extent free of its water coat and is spatially oriented in the correct way. At the same time, the enzyme limits the number of degrees of freedom, directing the reaction along a certain spatial coordinate.

ON THE ARRHENIUS AND EYRING EQUATIONS

The traditional ideas of the energy profile of enzymatic processes [1–4] are based on the phenomenological Arrhenius equation, which was proposed as early as the 19th century and on the theory of the transition activated complex proposed by G. Eyring in the middle of the 19th century. The Arrhenius equation includes the activation energy and temperature, which are exponentially related to the reaction rate. Eyring’s equation features thermodynamic parameters, Gibbs free energy, entropy, and enthalpy, and temperature, with these parameters being attributed to the substrate–enzyme transition complex [5]. At the heart of Eyring’s theory is the postulate that the Maxwell–Boltzman energy distribution is not violated in the course of an enzymatic reaction. In other words, nonequilibrium “hot” intermediate complexes or products should not occur in the active center of the enzyme whose temperature differs from the environmental temperature [4, 5].

The formal replacement of the Arrhenius activation energy with the Gibbs free energy that was made by Eyring and his co-authors [5] has no physical sense, since thermodynamics is well suited to finding the balance of the process, but not to describing the reaction pathway i.e., its mechanism. Using the Arrhenius equation and Eyring’s theory it appears impossible to adequately explain the decrease of the energy activation barrier by the enzyme.

EXCITED STATES

There is no experimental evidence for the special “active molecules” of Arrhenius and Eyring’s postulate of the equilibrium energy distribution during catalysis. On the contrary, nonequilibrium electron and oscillatory excited states detected by chemiluminescence [6], bioluminescence [7], and infrared radiation [8–10] have been found in many catalytic reactions. In the course of common enzymatic processes electron paramagnetic resonance signal is found [11, 12] in a number of cases, which indicates the presence of triplet excited states. In introducing an energy acceptor it becomes possible to detect the presence of such states by the energy emission of the acceptor, as, for example, in the case of peroxidase [13].

THE ACTIVE MOLECULE NUMBER

When the temperature in the environment rises by 10 degrees, the enzymatic reaction rate at least doubles [1, 2]. Arrhenius and subsequently, Eyring explained this by an increase in the portion of “active” molecules. This explanation is erroneous. As an example, at 300 and 310 K the reaction rate increases by 1.03 times, and not 2 times. This may be easily calculated when two Arrhenius equations are combined:

$${{{{a}_{1}}} \mathord{\left/ {\vphantom {{{{a}_{1}}} {{{a}_{2}}}}} \right. \kern-0em} {{{a}_{2}}}} = \exp \left\{ {{{E}_{{{\text{ta}}}}}{{\left( {{{T}_{2}} - {{T}_{1}}} \right)} \mathord{\left/ {\vphantom {{\left( {{{T}_{2}} - {{T}_{1}}} \right)} {\left( {R{{T}_{1}}{{T}_{2}}} \right)}}} \right. \kern-0em} {\left( {R{{T}_{1}}{{T}_{2}}} \right)}}} \right\},$$
(1)

where Eta is the true activation energy, a1 is the number of active molecules at the temperature T1, and a2 gives the number of active molecules at the increased temperature T2.

THE ROLE OF VISCOSITY

Actually, the increase in the rate of enzymatic activity with a certain rise in temperature is to a large extent due to a simple diffusion factor, namely, a decrease in the viscosity of the solvent [14, 15]. As an example, when the temperature increases from 10 to 30°C, the viscosity of water decreases from 1.3 to 0.8 poise [16]. In addition, at increased temperatures the microviscosity of the protein globule decreases while its conformational mobility increases [17]. Neither Arrhenius, nor Eyring took these issues into account.

ACTIVATION ENERGY

Placing the environmental temperature (T) into the Arrhenius equation and considering it constant during all the elementary stages of the enzymatic process is generally accepted. By measuring the dependence of the reaction rate on T, we can find some activation energy Ea. As was shown in [18], this Ea is an apparent value and has no direct relation to the true activation energy, that is, the energy required to trigger the reactions within the substrate-enzyme complex. The author of the [18] properly pointed to the incorrectness of the Arrhenius interpretation and the inapplicability of equilibrium thermodynamics to the description of enzymatic processes. The true activation energy is the energy required to overcome the activation barrier, i.e., to activate the substrate, while the apparent energy is the energy calculated using the Arrhenius equation based on the temperature–reaction rate dependence. The apparent activation energy is a result of diffusion and the viscosity of the medium [14], as well as the microviscosity of the enzyme itself.

ELECTRONIC AND NUCLEAR TRANSITIONS

Another shortcoming of Eyring’s theory is that the “reaction coordinate” is considered to be the distance between the nuclei, whereas the distance between the nuclei and electrons should be used, since it is the electronic orbits that interact [19]. Electron transitions occur instantly and nuclei then slowly adopt the new configuration [20]. The authors of [20] clearly indicated the role of excited states in enzymatic reactions and even estimated their contribution to some reactions, although they did not consider the processes to be nonequilibrium or that they take place at high intramolecular temperatures. The energy contribution of nuclear (oscillatory) movements is an order of magnitude smaller than that of the electronic movements. As well, electron-oscillatory transitions are quant-like and discrete, an idea that is fundamentally different from the concept of the continuous trajectory of motion in Eyring’s theory.

CHEMISORPTION ENERGY

The adsorption of the substrate on the enzyme leads to the emission of a large chemisorption energy quantum Echem [4, 20], which obviously exceeds Ea. This energy quantum is almost completely used so that the subsequent stages proceed without the need for activation. In the case of complete neutralization of the two opposite electric charges, Echem may be calculated according to the following equation:

$${{E}_{{{\text{chem}}}}} = 2.15 \times {{{{{10}}^{6}}N{{q}_{1}}{{q}_{2}}} \mathord{\left/ {\vphantom {{{{{10}}^{6}}N{{q}_{1}}{{q}_{2}}} {r\varepsilon }}} \right. \kern-0em} {r\varepsilon }},$$
(2)

where q1 and q2 are the positive and negative charges, N is Avogadro’s number, r is the final distance between the neutralized charges, and ε is dielectric permittivity (not of the environment, but of the substrate and active enzyme center groups). At r = 1.5 Å (covalent bond length) and ε = 1, the maximum Echem = 250 kcal/mol occurs. This energy is enough to break any covalent bond. ε ~ 1 may be used for the enzymes since water molecules (ε = 81) normally do not fit between the neutralized groups. In the case of incomplete charge neutralization (dipoles, etc.) as well as at ε > 1, Echem may be lower, although it still exceedes the activation energy. The charge neutralization formula may be used for certain enzymatic reactions when charge sizes, dipole moments, distances between charges, and the dielectric coefficient are known.

ELECTRON-OSCILLATORY EXCITED COMPLEXES

Echem is related to the enthalpy component ΔH of the Gibbs free energy ΔG. Not all Echem is used to produce the useful effect, since a small part of it is lost as heat. The adsorption heat ΔH (enthalpy), which is measured using calorimetry approaches, represents this small lost energy:

$$\Delta H = {{E}_{{{\text{chem}}}}} - E^{*},$$
(3)

where E* is the energy of the transition electron-oscillatory excited complex (EOEC). The loss of a small (oscillatory) part of energy as heat accompanying the substrate (S) adsorption onto the enzyme (E) leads to the irreversibility of the subsequent stages. However, the reversibility of the enzymatic reaction is preserved in a strict sense (the forward and the reverse pathways coincide):

$${\text{S}} + {\text{E}} = {\text{EOEC}} \cdot {\text{SE}} = {\text{P}} + {\text{E}}.$$
(4)

The formed EOEC quickly reorganizes and then transforms into the product (P). This happens before the EOEC is able to relax into the equilibrium state. The EOEC energy ultimately makes it possible for the subsequent stages including regrouping, electron or group transfer, and desorption of the reaction product, to be completed.

Volkenstein et al. [20] used the idea that in the enzyme’s active center, the molecular orbits of the basic and excited states of the substrate mix, thus changing its geometry, dipole moment, and pK to describe the energy profile of the enzymatic reaction.

It should be noted that the absence of light emission during most enzymatic reactions does not mean the absence of an EOEC. In such reactions, the energy of the EOEC is not emitted (in the ultraviolet or visible spectrum), but is rather accumulated in the product and is partially lost as thermal infrared radiation absorbed by water.

THE LOCAL DISEQUILIBRIUM TEMPERATURE

Enzymatic reactions are neither equilibrium, nor isothermal, since electronic (with the energies of approximately 100 kcal/mol) and/or oscillatory (with energies of approximately 10 kcal/mol) degrees of freedom are excited in their course.

The enzyme reduces the activation barrier by producing the EOEC. During all stages the Maxwell–Boltzmann energy distribution is strongly violated and the local (nonequilibrium) temperature in the active center increases. Reaction with the participation of an enzyme always proceed at a high nonequilibrium intramolecular temperature. The local temperature of the substrate and that of the active center coincide with the environmental temperature only before and after the catalytic act, but they strongly do not coincide during the catalytic act.

It should be especially noted that the widespread idea that temperature is a measure of average kinetic energy is far from being complete. This is true for gases where intermolecular interactions are negligible and electronic and oscillatory levels are not occupied. In fact, temperature is a measure of the occupation of all a molecules’ degrees of freedom, i.e., rotational, translational, oscillatory, and electronic [15, 21]. As an example, in the case of photo-excitation of molecules in the ultraviolet and visible range the main contribution is made by electronic transitions that correspond to nonequilibrium temperatures of thousands of degrees [17, 21].

TEMPERATURE, ENERGY, WAVE NUMBER, AND FREQUENCY

At the temperature of 300 K, the average energy of the molecules is known to be as low as 0.6 kcal/mol [15], i.e. it corresponds to the energy of translational and rotational degrees of freedom. According to the wave number scale, 0.6 kcal/mol corresponds to 200 cm–1, i.e., to the far infrared range, and according to the frequency scale, it corresponds to 10–12 s. If we assume υ = 2000 cm–1 (6 kcal/mol), it will correspond to 10–10 s according to the frequency scale, i.e., to oscillatory transitions. If we assume υ = 20 000 cm–1 (60 kcal/mol), then we obtain 10–9, which is the typical life time of the electron-excited state [17]. It is also the time that is required for an elementary act of enzymatic catalysis to occur, while the rate of the entire enzymatic reaction is limited by its slowest stage, namely, product desorption [1, 3, 17].

Using the simple Wien formula for the radiation spectrum at a certain temperature [21] (or more precisely, the Planck’s formula) and given the energy of the emitted quanta (E = hυ), we can find the nonequilibrium temperature (E = kT). Electronic transitions correspond to nonequilibrium local temperatures of several thousand degrees. As an example, the nonequilibrium temperature is approximately 5000 K for bioluminescence in the yellow spectral band.

CONCLUSIONS

In all common “dark” enzymatic reactions (and not only in bioluminescent and photochemical reactions), the active center is strongly “heated” by the substrate, which facilitates not only the energy component, but also the steric component, as heating increases the dynamics and mobility of the active center and facilitates product desorption.

The rate of the enzymatic reaction in a solution is limited by diffusion and steric factors; however, the elementary stages of catalysis are essentially monomolecular, in a similar manner to the known thermochemical reactions that do not obey Eyring’s theory because they follow a non-isothermal path and are accompanied by the release of energy quanta of 10–50 kcal/mol [22, 23].

The structural changes at different stages of enzymatic reactions can be described using molecular dynamics methods [24]. The energy profile usually includes the stages with energies of 5–20 kcal/mol, which are significantly higher than the equilibrium ones (0.6 kcal/mol). As noted in [24], the results of calculations using equilibrium molecular dynamics methods often do not coincide with the experimental data.