Reformulated Kinetics of Immobilized Enzymes in Non-Conventional Media: A Case of Lipase-Catalyzed Esterification

Abstract

Several approaches have been reported that aim to achieve simplified standardizations of the kinetic behavior of immobilized enzymes under specific experimental conditions. We have previously published simplified rate equations based on the kinetics of immobilized enzymes. Recently, new experimental results have become available on the kinetics and mechanisms of esterifications catalyzed by immobilized lipase in unconventional media, and consequently, a reformulation of their kinetics is necessary. In this work, we report the development of simplified rate equations relating the aforementioned reaction conditions on a new basis, considering our kinetic and mechanistic results. We provide experimental evidence that two different mechanisms describe the esterifications catalyzed by immobilized lipase, either in anhydrous organic solvent (n-hexane) or under non-solvent conditions. A ping-pong bi–bi mechanism with double dead-end substrate inhibition by both the fatty acid and the alcohol has been found to apply in the former case, while in the latter case the esterification proceeds via an ordered bi–bi mechanism with single dead-end substrate inhibition by ethanol. This study may be biotechnologically useful, as the increased use of immobilized enzymes, whether in academic research or in industry, requires sustainable development of new and environmentally friendly synthetic processes.

1. Introduction

Over the years, many lipase-catalyzed esterification methods for fatty acids and alcohols have become more productive [1]. The success of these methods is based on the ever-increasing understanding of the corresponding enzymatic reaction mechanisms [2] either in nonpolar organic solvents or in solvent-free systems, under anhydrous conditions and using immobilized lipases [3,4,5,6,7]. Fatty acids (up to C-10), which exhibit similar pK_a values in aqueous media, have been esterified with the corresponding alcohols mainly under anhydrous reaction conditions [3,4,5,6,7]. It has been experimentally shown that esterifications catalyzed by immobilized lipase proceed through similar mechanisms under the aforementioned anhydrous conditions. Furthermore, relevant techniques and/or probes have been applied as key tools (e.g., use of appropriately immobilized lipases, experimental designs, control of diffusion effects, solvent isotope effects, proton inventories, etc.) [3,5,6,8] to promote newer, efficient and beneficial esterification methods. Subsequently, a remarkable, economical and environmentally friendly construction of building blocks was achieved, providing esters (food additives) of high purity through the potential exploitation of various wastes, to a large extent, from food industries [5,6]. It seems more likely that knowledge of the exact kinetic mechanism is always beneficial when complex enzyme-catalyzed reactions are used to synthesize high-quality industrial products [8,9,10,11]. Under certain conditions, the experimenter faces different catalytic function for immobilized enzymes. Moreover, it should be emphasized that immobilized enzymes apparently deviate from conventional Michaelis–Menten kinetics, depending on the reaction conditions. The heterogeneity of the reaction systems and/or the complete absence of water can negatively affect the enzyme molecule due to potential structural changes. Usually, in these cases, multiparametric and difficult-to-solve model equations of two independent variables describe the process under consideration, making the fitting of experimental data questionable. Therefore, to fit equations such as these to experimental responses, it is necessary to apply surface fitting of experimental kinetic data, a difficult task. Consequently, the use of apparent Michaelis–Menten parameters based on both experimental and theoretical approaches [8,12,13] is welcome, as it offers quick and reliable information on the magnitude of crucial kinetic parameters, which are important and useful in biotechnology and industrial applications. The reader should feel no dilemma. His/her choice depends on the relevant needs and perspectives. Academic researchers may prefer to know the magnitudes of all parameters of a mechanistic model equation, whereas scientists involved in either biotechnology or industrial applications may choose to know the essential and required parameters.

This work reports on a reformulation of esterification kinetics catalyzed by immobilized Candida antarctica lipase B (Novozym 435, CALB). It is based on our previous studies [3,5,6] reporting experimentally proven mechanisms of esterification of short-chain fatty acids (propionate and butyrate) with alcohols (ethanol and n-butanol), catalyzed by immobilized lipase in non-conventional anhydrous media (i.e., ping-pong bi–bi and ordered bi–bi mechanisms, involving double dead-end and single dead-end substrate inhibition, respectively). In addition, we report here a formulation of simplified rate equations that coincide with the experimental conditions of the above esterifications, based on the new concepts we have previously established [12].

2. Results and Discussion

2.1. General Assessment

It seems reasonable and, in a sense, satisfactory and sufficient to adopt the concept of bypassing complex procedures such as surface fitting of experimental kinetic data by introducing reasonable simplifications of multiparametric and multivariable equations [7,11,13]. Undoubtedly, the evaluation of the functionality and adequacy of any suggested approximation should ensure that it meets the accuracy, validity and suitability requirements of the cases in which it will be applied. All the specific equations formulated to describe various cases of esterification processes (Appendix A: Equations (A1)–(A4)) were successfully transformed into simplified forms involving only two parameters, namely, k_cat and ${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}$. All transformations correspond to the reaction conditions and were found to be successful regardless of the physical significance of ${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}$. Diffusional effects were not taken into account in this work because, in all cases of esterifications, the formed water molecules were continuously removed from the reaction mixtures. This is clearly seen in the formulation of Equations (A1)–(A4) (in the Appendix A) [3,5,6]. Furthermore, it was shown that the investigated cases can practically be described by the same equations for both mentioned mechanisms. Our results are annotated based on the acceptable case-by-case conditions we considered, which have been coded accordingly.

2.2. The Specific Cases [I] and [II]

2.2.1. Subcases [A]-[i]

The apparent K_m is described by the relation ${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}=K_{\mathrm{m}{\mathrm{S}}_1}+K_{\mathrm{m}{\mathrm{S}}_2}$. The elimination of term ${\displaystyle \frac{{{\displaystyle k}}_{-1}}{{{\displaystyle k}}_1}}{K}_{\mathrm{m}{\mathrm{S}}_2}$ in [II]-[A]-[i] is explained thoroughly in the text.

2.2.2. Subcases [A]-[ii]

The apparent K_m is given by the relation ${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}={\displaystyle \frac{{{\displaystyle [\mathrm{S}]}}_{\mathrm{t}}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_1}}}(K_{\mathrm{m}{\mathrm{S}}_1}+K_{\mathrm{m}{\mathrm{S}}_2})$; it differs compared to that in cases [i], and obviously the value of this ${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}$ is higher than that of the previous one, as it is multiplied by a factor >1. This may be due to the fact, that at higher [S₂]_t concentrations, the inhibition of alcohol is more pronounced than in the former case.

2.2.3. Subcases [A]-[iii]

In these cases, the formula for ${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}$ is described by a seemingly more complicated relation, ${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}={\displaystyle \frac{{{\displaystyle K}}_{{{\displaystyle \mathrm{m}\mathrm{S}}}_1}{\displaystyle {{\displaystyle [{{\displaystyle \mathrm{S}}}_2]}}_{\mathrm{t}}^2}{{\displaystyle +K}}_{{{\displaystyle \mathrm{m}\mathrm{S}}}_2}{{\displaystyle [{{\displaystyle \mathrm{S}}}_1]}}_{\mathrm{t}}^2}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_1}{{\displaystyle [{{\displaystyle \mathrm{S}}}_2]}}_{\mathrm{t}}}}$. More details on this formulation are given in the text, within Remark 1.

2.2.4. Subcases [B]-[i]

These cases are similar to those of [I] and [II] [A]-[i]; they display the same relation for the parameter ${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}$.

2.2.5. Subcases [B]-[ii]

These cases seem to be different from those in [I] and [II] [A]-[i]. The parameter ${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}$ is described by the relation ${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}={\displaystyle \frac{K_{\mathrm{m}{\mathrm{S}}_1}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_2}}}{\displaystyle {{\displaystyle [\mathrm{S}]}}_{\mathrm{t}}}+K_{\mathrm{m}{\mathrm{S}}_2}$, which differs, but it is just as simple as that found in cases [I] and [II] [A]-[ii]. Additionally, in the current cases, only a single dead-end substrate (alcohol) inhibition has been taken into account.

2.2.6. Subcases [B]-[iii]

Remark 2 describes sufficiently the formula for the parameter ${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}$ in the current cases, i.e., ${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}={\displaystyle \frac{K_{\mathrm{m}{\mathrm{S}}_1}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_2}}}+K_{\mathrm{m}{\mathrm{S}}_2}{\displaystyle \frac{{{\displaystyle [{{\displaystyle \mathrm{S}}}_1]}}_{\mathrm{t}}}{{{\displaystyle [{{\displaystyle \mathrm{S}}}_2]}}_{\mathrm{t}}}}$; the necessary assumptions and processing are based on the fact that the relations ${\displaystyle \frac{{{\displaystyle [{{\displaystyle \mathrm{S}}}_2]}}_{\mathrm{t}}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_2}}}>>1$ and [S₁]_t < [S₂]_t both hold true.

2.3. Validation of the Applied Approximations

Approximations may include errors and vary depending on the experimental conditions. However, what counts is the successful satisfaction of the needs and perspectives of the user of any proposed standardization as a result of the applied approximations. Finally, and according to Scheme 1, as well as based on our published results [5,6], it is important to emphasize that the constant k₄_ppg rate-limits the esterification when a ping-pong bi–bi mechanism is in effect, which involves either double or single dead-end substrate inhibition.

Therefore, the relation k₂_ppg >> k₄_ppg is valid, showing also that the relation k_cat ≈ k₄_ppg holds true. Furthermore, by definition and by considering also our previous findings, we show that the former relation holds true according to $k_{\mathrm{c}\mathrm{a}\mathrm{t}}={\displaystyle \frac{k_2 k_4}{k_2+k_4}}\approx {\displaystyle \frac{k_2 k_4}{k_2}}=k_4$. Consequently, the dependencies of parameters K_mS1 and K_mS2 are $K_{\mathrm{m}{\mathrm{S}}_1}={\displaystyle \frac{k_4(k_{-1}+k_2)}{k_1(k_2+k_4)}}\approx {\displaystyle \frac{k_4{k}_2}{k_1{k}_2}}={\displaystyle \frac{k_4}{k_1}}$ and $K_{\mathrm{m}{\mathrm{S}}_2}={\displaystyle \frac{k_2(k_{-3}+k_4)}{k_3(k_2+k_4)}}\approx {\displaystyle \frac{k_2{k}_4}{k_3{k}_2}}={\displaystyle \frac{k_4}{k_3}}$, respectively. These relations strengthen the simplifications that are proposed in this work, indicating that the parameters $K_{\mathrm{m}{\mathrm{S}}_1}$ and $K_{\mathrm{m}{\mathrm{S}}_2}$ can be expressed with simpler relations. For the sake of simplicity, the index _ppg has been omitted from all the above rate constants.

3. Methods and Tools

3.1. Summary

Various methods have been applied that provide realistic approximations, aiming to facilitate decision-making to distinguish the optimum among various enzymatic biotechnological synthetic and other processes. In this study, simplifying techniques for fitting multiparametric and multivariable rate equations to experimental responses resulting from esterifications catalyzed by immobilized lipase are developed and presented. The sources of the necessary experimental data and mechanistic information were our newly published results reporting esterifications of fatty acids and short-chain alcohols catalyzed by immobilized lipase in anhydrous organic solvent and/or under anhydrous solvent-free conditions [3,5,6].

3.2. The Subject Matter

We relied on the reported experimental data concerning the esterification reactions of short-chain fatty acids and alcohols, catalyzed by immobilized lipase (CALB) in anhydrous organic solvents and/or in anhydrous solvent-free media [2,3,4,5,6,8]. Relevant theoretical works and computational approaches were also considered [14,15].

According to the experimental data, it has been shown that the aforementioned esterifications, which are catalyzed by immobilized lipase (CALB) in anhydrous n-hexane under continuous withdrawal of the produced water molecules, proceed through a ping-pong bi–bi reaction scheme followed by double dead-end substrate inhibition. On the other hand, an ordered bi–bi reaction scheme followed by single dead-end substrate (alcohol) inhibition was identified when the esterifications were performed under similar conditions but in a solvent-free medium (in anhydrous alcohol) [3,5,6]. Scheme 1 shows both ping-pong bi–bi and ordered bi–bi mechanisms [5,6,16]. The indices _ppg and _ord refer to the ping-pong bi–bi mechanism and ordered bi–bi mechanism, respectively.

3.3. Useful Definitions and Approximations

The reported and accepted rate equations in all cases of both ping-pong bi–bi and ordered bi–bi mechanisms are known (see Appendix A) [4,5,6,8,12,16,17]. For the sake of quickly referencing the parameters of these multiparametric and multivariable (two independent variables) rate equations, the following is valid:

k_cat is the known catalytic constant, also referred to as the turnover number. Reasonable approximations for these parameters as well as for the two aforementioned mechanisms can be validated/confirmed based on the reaction conditions considered in this work, also taking into account both the estimated values of all parameters and the results from the kinetic studies of isotope effects and proton inventories [5,6].

$$k_{\mathrm{c}\mathrm{a}\mathrm{t}}={\displaystyle \frac{k_2 k_4}{k_2+k_4}}\approx {\displaystyle \frac{k_2 k_4}{k_2}}=k_4$$

The same parameter for the ordered bi–bi mechanism with single dead-end substrate (alcohol) inhibition can be approximated as $k_{\mathrm{c}\mathrm{a}\mathrm{t}}={\displaystyle \frac{k_3 k_4}{k_3+k_{-3}+k_4}}\approx {\displaystyle \frac{k_3 k_4}{k_3+k_4}}$ [5,6], which holds true due to the validity of the relations k_{4_ppg} << k_{2_ppg} and k_{4_ppg} < k_{3_ppg}, as well as the fact that the values of both k_{-1_ppg} and k_{-3_ppg} approach zero. For simplicity purposes, the indices _ppg and _ord have been omitted from the previous relations.

K_mS1 and K_mS2 are the Michaelis–Menten-like parameters related to the reactions of the corresponding enzymatic species with either substrate S₁ (the acid) or substrate S₂ (the alcohol). Based on the same findings, as referred to in the case of the k_cat parameter, similar reasonable and interesting approximations can be applied for both the K_mS1 and K_mS2 parameters. Dealing with the ping-pong bi–bi mechanism, comprising double-dead-end substrate inhibition, these parameters can be approximated as $K_{\mathrm{m}{\mathrm{S}}_1}={\displaystyle \frac{k_4(k_{-1}+k_2)}{k_1(k_2+k_4)}}\approx {\displaystyle \frac{k_4}{k_1}}$ and $K_{\mathrm{m}{\mathrm{S}}_2}={\displaystyle \frac{k_2(k_{-3}+k_4)}{k_3(k_2+k_4)}}\approx {\displaystyle \frac{k_2{k}_4}{k_3{k}_2}}={\displaystyle \frac{k_4}{k_3}}$, respectively. The corresponding parameters for the ordered bi–bi mechanism with single-dead-end substrate (alcohol) inhibition can be approximated as

$$K_{\mathrm{m}{\mathrm{S}}_1}={\displaystyle \frac{k_2{k}_3{k}_{-\mathrm{i}}{k}_4+k_{-1}{k}_3{k}_{\mathrm{i}}{k}_4+k_{-1}{k}_{-2}{k}_{\mathrm{i}}{k}_{-3}+k_{-1}{k}_{-2}{k}_{\mathrm{i}}{k}_4}{k_1{k}_2{k}_{-\mathrm{i}}(k_3+k_{-3}+k_4)}}\approx {\displaystyle \frac{k_3{k}_4}{k_1(k_3+k_4)}}$$

$$K_{\mathrm{m}{\mathrm{S}}_2}={\displaystyle \frac{k_{-2}{k}_4+k_{-2}{k}_3+k_3{k}_4}{k_2(k_3+k_{-3}+k_4)}}={\displaystyle \frac{k_3{k}_4}{k_2(k_3+k_4)}}$$

Accordingly, and for the purpose of simplicity, the _ppg and _ord indices have also been omitted from the relations of the dependencies of parameters K_mS1 and K_mS2. The obvious relation that is valid in both mechanisms under consideration, such as K_mS1 > K_mS2, argues in favor of these approximations. The release of the product H₂O in the ping-pong bi–bi mechanism, which involves double dead-end substrate inhibition, follows the formation of the ES₁ complex and precedes the binding of the S₂ (alcohol) substrate to the acyl-enzyme species Eacyl_S1. In that case, ester formation is the rate-limiting step of this mechanism.

Furthermore, the formed H₂O is continuously removed from the esterification medium, and the rate constant k_{4_ppg} is found to be sensitive to isotopic (proton) substitution [5]. These arguments support the approximations that the values of both k_-1, and k_{-3_ppg} approach zero. Furthermore, considering the ordered bi–bi mechanism comprising single dead-end substrate inhibition, S₂ binds to the ES₁ complex. Then, two consecutive reversible steps follow before the irreversible one leading to the simultaneous formation of both H₂O and the ester. In this case too, the formed H₂O is continuously removed from the esterification medium; additionally, the rate constant k_{3_ord} is found to be sensitive to isotopic (proton) substitution [5,6]. These latter findings support the claim that the rate constant k_{4_ord} partially rate-limits this mechanism (ordered bi–bi), as it depends on the previous step (reversible and/or irreversible). These arguments support the approximations that the values of both k_{-2_ord} and k_{-3_ord} approach zero [3,18,19,20].

[E]_t represents the total concentration of the immobilized lipase enzyme (CALB), which participates in the esterification reactions, whereas [Eacyl_S1] is the steady-state concentration of the enzymatic species designated as acyl-enzyme. K_iS1 and K_iS2 are the dissociation constants of the Eacyl_S1S₂ and ES₂ species, respectively. [S₁]_t, [S₁], [S₂]_t and [S₂] represent the total concentrations of the substrates and the concentrations of free substrates, respectively (S₁ for acid and S₂ for alcohol) participating in the esterification reactions.

3.4. The Development of Simplified Rate Equations

We considered the available and relatively recently published estimated values of the abovementioned parameters [5,6], which were obtained from esterifications catalyzed by immobilized lipase (CALB) in both anhydrous organic solvents and non-solvent conditions. Our previous kinetic and mechanistic results, as well as our experience, were also taken into account [12]. Consequently, and based on the above, we can make the following realistic assumptions and considerations:

(a)

It is always valid that K_iS1 ≈ K_iS2 and that the value of ${\displaystyle \frac{{{\displaystyle k}}_{-1}}{{{\displaystyle k}}_1}}$ × K_mS2 approaches zero.

(b)

When [S₁]_t ≈ [S₂]_t and [S₁]_t, [S₂]_t >> K_mS1, K_mS2, it holds true that ${\displaystyle \frac{{{\displaystyle [{\mathrm{S}}_1]}}_{\mathrm{t}}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_1}}}\approx {\displaystyle \frac{{{\displaystyle [{\mathrm{S}}_2]}}_{\mathrm{t}}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_2}}}>>1$, whereas when [S1]t ≠ [S2]t, it holds true that ${\displaystyle \frac{{{\displaystyle [{\mathrm{S}}_1]}}_{\mathrm{t}}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_1}}}>>1$, and ${\displaystyle \frac{{{\displaystyle [{\mathrm{S}}_2]}}_{\mathrm{t}}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_2}}}>>1$.

(c)

In cases when [S₁]_t, [S₂]_t << K_mS1, K_mS2 the opposite relations hold true, i.e., for [S₁]_t ≈ [S₂]_t, it is valid that ${\displaystyle \frac{{{\displaystyle [{\mathrm{S}}_1]}}_{\mathrm{t}}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_1}}}\approx {\displaystyle \frac{{{\displaystyle [{\mathrm{S}}_2]}}_{\mathrm{t}}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_2}}}<<1$, whereas for [S₁]_t ≠ [S₂]_t, the relations and ${\displaystyle \frac{{{\displaystyle [{{\displaystyle \mathrm{S}}}_2]}}_{\mathrm{t}}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_2}}}<<1$ are valid.

[I]

Specific cases of esterifications following a ping-pong bi–bi mechanism:

[A]

With double dead-end substrate inhibition:

[i]

If the accepted conditions are that [S₁]_t ≈ [S₂]_t << K_mS1 and K_mS2, then the following is true:

$$\begin{array}{ll}\mathrm{v}=& {\displaystyle \frac{{[\mathrm{E}]}_{\mathrm{t}}{k}_{\mathrm{c}\mathrm{a}\mathrm{t}}[\mathrm{S}]}{K_{\mathrm{m}{\mathrm{S}}_1}(1+\frac{[\mathrm{S}]}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_2}})+K_{\mathrm{m}{\mathrm{S}}_2}(1+\frac{[\mathrm{S}]}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_1}})+[\mathrm{S}]}}\approx {\displaystyle \frac{{[\mathrm{E}]}_{\mathrm{t}}{k}_{\mathrm{c}\mathrm{a}\mathrm{t}}[\mathrm{S}]}{(1+\frac{[\mathrm{S}]}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_1}}) (K_{\mathrm{m}{\mathrm{S}}_1}+K_{\mathrm{m}{\mathrm{S}}_2})+[\mathrm{S}]}}\approx \\ & {\displaystyle \frac{{[\mathrm{E}]}_{\mathrm{t}}{k}_{\mathrm{c}\mathrm{a}\mathrm{t}}[\mathrm{S}]}{(K_{\mathrm{m}{\mathrm{S}}_1}+K_{\mathrm{m}{\mathrm{S}}_2})+[\mathrm{S}]}} \Rightarrow \mathrm{v}={\displaystyle \frac{{[\mathrm{E}]}_{\mathrm{t}}{k}_{\mathrm{c}\mathrm{a}\mathrm{t}}[\mathrm{S}]}{{{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}+[\mathrm{S}]}}, \end{array}$$

where ${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}=K_{\mathrm{m}{\mathrm{S}}_1}+K_{\mathrm{m}{\mathrm{S}}_2}$.

[ii]

If the accepted conditions are that [S₁]_t ≈ [S₂]_t >> K_mS1 and K_mS2, then, according to (i) above, the following is true:

$$\mathrm{v}\approx {\displaystyle \frac{{[\mathrm{E}]}_{\mathrm{t}}{k}_{\mathrm{c}\mathrm{a}\mathrm{t}}[\mathrm{S}]}{\frac{{{\displaystyle [\mathrm{S}]}}_{\mathrm{t}}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_1}}(K_{\mathrm{m}{\mathrm{S}}_1}+K_{\mathrm{m}{\mathrm{S}}_2})+[\mathrm{S}]}} \Rightarrow \mathrm{v}={\displaystyle \frac{{[\mathrm{E}]}_{\mathrm{t}}{k}_{\mathrm{c}\mathrm{a}\mathrm{t}}[\mathrm{S}]}{{{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}+[\mathrm{S}]}},$$

where ${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}={\displaystyle \frac{{{\displaystyle [\mathrm{S}]}}_{\mathrm{t}}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_1}}}(K_{\mathrm{m}{\mathrm{S}}_1}+K_{\mathrm{m}{\mathrm{S}}_2})$.

[iii]

If the accepted conditions are that [S₁]_t ≠ [S₂]_t >> K_mS1 and K_mS2, then, according to the assumptions, the following is true:

$$\begin{array}{ll}\mathrm{v}=& {\displaystyle \frac{{[\mathrm{E}]}_{\mathrm{t}}{k}_{\mathrm{c}\mathrm{a}\mathrm{t}}[{\displaystyle \mathrm{S}}]}{K_{\mathrm{m}{\mathrm{S}}_1}\frac{[{\displaystyle \mathrm{S}}]}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_2}}+K_{\mathrm{m}{\mathrm{S}}_2}\frac{[{\displaystyle \mathrm{S}}]}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_1}}+[{\displaystyle \mathrm{S}}]}}\approx {\displaystyle \frac{{[\mathrm{E}]}_{\mathrm{t}}{k}_{\mathrm{c}\mathrm{a}\mathrm{t}}[{\displaystyle \mathrm{S}}]}{\frac{{{\displaystyle K}}_{{{\displaystyle \mathrm{m}\mathrm{S}}}_1} {{\displaystyle [{{\displaystyle \mathrm{S}}}_2]}}^{{\displaystyle 2}}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_2}}+\frac{{{\displaystyle K}}_{{{\displaystyle \mathrm{m}\mathrm{S}}}_2} {{\displaystyle [{{\displaystyle \mathrm{S}}}_1]}}^{{\displaystyle 2}}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_1}}+[{{\displaystyle \mathrm{S}}}_1][{{\displaystyle \mathrm{S}}}_2]}}=\\ & {\displaystyle \frac{{[\mathrm{E}]}_{\mathrm{t}}{k}_{\mathrm{c}\mathrm{a}\mathrm{t}}[{\displaystyle \mathrm{S}}]}{\frac{{{\displaystyle K}}_{{{\displaystyle \mathrm{m}\mathrm{S}}}_1}[{{\displaystyle \mathrm{S}}}_2] }{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_2}}+\frac{{{\displaystyle K}}_{{{\displaystyle \mathrm{m}\mathrm{S}}}_2} }{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_1}}\frac{{{\displaystyle [{{\displaystyle \mathrm{S}}}_1]}}^{{\displaystyle 2}}}{[{{\displaystyle \mathrm{S}}}_2]}+[{{\displaystyle \mathrm{S}}}_1]}} \Rightarrow \mathrm{v}={\displaystyle \frac{{[\mathrm{E}]}_{\mathrm{t}}{k}_{\mathrm{c}\mathrm{a}\mathrm{t}}[{{\displaystyle \mathrm{S}}}_1]}{{{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}+[{{\displaystyle \mathrm{S}}}_1]}},\end{array}$$

where ${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}={\displaystyle \frac{{{\displaystyle K}}_{{{\displaystyle \mathrm{m}\mathrm{S}}}_1}{\displaystyle {{\displaystyle [{{\displaystyle \mathrm{S}}}_2]}}_{\mathrm{t}}^2}{{\displaystyle +K}}_{{{\displaystyle \mathrm{m}\mathrm{S}}}_2{{\displaystyle [{\mathrm{S}}_1]}}_{\mathrm{t}}^2}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_1}{{\displaystyle [{{\displaystyle \mathrm{S}}}_2]}}_{\mathrm{t}}}}$.

Remark 1. 

In this part, we considered the assumptions that${\displaystyle \frac{{{\displaystyle [{{\displaystyle \mathrm{S}}}_1]}}_{\mathrm{t}}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_1}}}>>1, {\displaystyle \frac{{{\displaystyle [{{\displaystyle \mathrm{S}}}_2]}}_{\mathrm{t}}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_2}}}>>1$, and [S₁]_t < [S₂]_t; accordingly, [S₁] was chosen as the independent variable. If [S₁]_t > [S₂]_t is valid, then [S₂] should be chosen as the independent variable. By the index t it is denoted that in the value of the ${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}$, and under these conditions, the total concentrations both of substrates should be considered.

[B]

With single dead-end substrate (alcohol) inhibition:

[i]

If the accepted conditions are that [S₁]_t ≈ [S₂]_t << K_mS1 and K_mS2, while [S₁] and [S₂] ≠ 0, then the following is true:

$$\begin{array}{ll}\mathrm{v}=& {\displaystyle \frac{{[\mathrm{E}]}_{\mathrm{t}}{k}_{\mathrm{c}\mathrm{a}\mathrm{t}}[{{\displaystyle \mathrm{S}}}_1][{{\displaystyle \mathrm{S}}}_2]}{K_{\mathrm{m}{\mathrm{S}}_1}[{{\displaystyle \mathrm{S}}}_2](1+\frac{[{{\displaystyle \mathrm{S}}}_2]}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_2}})+K_{\mathrm{m}{\mathrm{S}}_2}[{{\displaystyle \mathrm{S}}}_1]+[{{\displaystyle \mathrm{S}}}_1][{{\displaystyle \mathrm{S}}}_2]}}\approx {\displaystyle \frac{{[\mathrm{E}]}_{\mathrm{t}}{k}_{\mathrm{c}\mathrm{a}\mathrm{t}}{[\mathrm{S}]}^2}{ K_{\mathrm{m}{\mathrm{S}}_1}[\mathrm{S}]+K_{\mathrm{m}{\mathrm{S}}_2}[\mathrm{S}]+{[\mathrm{S}]}^2}}=\\ & {\displaystyle \frac{{[\mathrm{E}]}_{\mathrm{t}}{k}_{\mathrm{c}\mathrm{a}\mathrm{t}}[\mathrm{S}]}{ K_{\mathrm{m}{\mathrm{S}}_1}+K_{\mathrm{m}{\mathrm{S}}_2}+[\mathrm{S}]}} \Rightarrow \mathrm{v}={\displaystyle \frac{{[\mathrm{E}]}_{\mathrm{t}}{k}_{\mathrm{c}\mathrm{a}\mathrm{t}}[{\displaystyle \mathrm{S}}]}{{{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}+[{\displaystyle \mathrm{S}}]}},\end{array}$$

where ${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}=K_{\mathrm{m}{\mathrm{S}}_1}+K_{\mathrm{m}{\mathrm{S}}_2}$.

[ii]

If the accepted conditions are that [S₁]_t ≈ [S₂]_t >> K_mS1 and K_mS2, while [S₁] and [S₂] ≠ 0, then the following is true:

$$\begin{array}{ll}\mathrm{v}=& {\displaystyle \frac{{[\mathrm{E}]}_{\mathrm{t}}{k}_{\mathrm{c}\mathrm{a}\mathrm{t}}[{{\displaystyle \mathrm{S}}}_1][{{\displaystyle \mathrm{S}}}_2]}{K_{\mathrm{m}{\mathrm{S}}_1}[{{\displaystyle \mathrm{S}}}_2](1+\frac{[{{\displaystyle \mathrm{S}}}_2]}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_2}})+K_{\mathrm{m}{\mathrm{S}}_2}[{{\displaystyle \mathrm{S}}}_1]+[{{\displaystyle \mathrm{S}}}_1][{{\displaystyle \mathrm{S}}}_2]}}\approx {\displaystyle \frac{{[\mathrm{E}]}_{\mathrm{t}}{k}_{\mathrm{c}\mathrm{a}\mathrm{t}}{[\mathrm{S}]}^2}{ \frac{K_{\mathrm{m}\mathrm{S}1}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_2}}{[\mathrm{S}]}^2+K_{\mathrm{m}{\mathrm{S}}_2}[\mathrm{S}]+{[\mathrm{S}]}^2}}=\\ & {\displaystyle \frac{{[\mathrm{E}]}_{\mathrm{t}}{k}_{\mathrm{c}\mathrm{a}\mathrm{t}}[\mathrm{S}]}{ \frac{K_{\mathrm{m}\mathrm{S}1}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_2}}{\displaystyle {{\displaystyle [{\displaystyle \mathrm{S}}]}}_{\mathrm{t}}}+K_{\mathrm{m}{\mathrm{S}}_2}+[\mathrm{S}]}} \Rightarrow \mathrm{v}={\displaystyle \frac{{[\mathrm{E}]}_{\mathrm{t}}{k}_{\mathrm{c}\mathrm{a}\mathrm{t}}[{\displaystyle \mathrm{S}}]}{{{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}+[{\displaystyle \mathrm{S}}]}},\end{array}$$

where ${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}={\displaystyle \frac{K_{\mathrm{m}{\mathrm{S}}_1}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_2}}}{\displaystyle {{\displaystyle [{\displaystyle \mathrm{S}}]}}_{\mathrm{t}}}+K_{\mathrm{m}{\mathrm{S}}_2}$.

[iii]

If the accepted conditions are that [S₁]_t ≠ [S₂]_t >> K_mS1 and K_mS2 while [S₁] and [S₂] ≠ 0, then the following is true:

$$\begin{array}{ll}\mathrm{v}=& {\displaystyle \frac{{[\mathrm{E}]}_{\mathrm{t}}{k}_{\mathrm{c}\mathrm{a}\mathrm{t}}[{{\displaystyle \mathrm{S}}}_1][{{\displaystyle \mathrm{S}}}_2]}{K_{\mathrm{m}{\mathrm{S}}_1}[{{\displaystyle \mathrm{S}}}_2](1+\frac{[{{\displaystyle \mathrm{S}}}_2]}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_2}})+K_{\mathrm{m}{\mathrm{S}}_2}[{{\displaystyle \mathrm{S}}}_1]+[{{\displaystyle \mathrm{S}}}_1][{{\displaystyle \mathrm{S}}}_2]}}\approx {\displaystyle \frac{{[\mathrm{E}]}_{\mathrm{t}}{k}_{\mathrm{c}\mathrm{a}\mathrm{t}}[{{\displaystyle \mathrm{S}}}_1]}{ \frac{K_{\mathrm{m}\mathrm{S}1}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_2}}+K_{\mathrm{m}{\mathrm{S}}_2}\frac{[{{\displaystyle \mathrm{S}}}_1]}{[{{\displaystyle \mathrm{S}}}_2]}+[{{\displaystyle \mathrm{S}}}_1]}}\\ & \Rightarrow \mathrm{v}={\displaystyle \frac{{[\mathrm{E}]}_{\mathrm{t}}{k}_{\mathrm{c}\mathrm{a}\mathrm{t}}[{\displaystyle \mathrm{S}}]}{{{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}+[{\displaystyle \mathrm{S}}]}},\end{array}$$

where ${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}={\displaystyle \frac{K_{\mathrm{m}{\mathrm{S}}_1}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_2}}}+K_{\mathrm{m}{\mathrm{S}}_2}{\displaystyle \frac{{{\displaystyle [{{\displaystyle \mathrm{S}}}_1]}}_{\mathrm{t}}}{{{\displaystyle [{{\displaystyle \mathrm{S}}}_2]}}_{\mathrm{t}}}}$.

Remark 2. 

It is quite similar to [I]-[A]-[iii]. In this part, we considered the assumptions that${\displaystyle \frac{{{\displaystyle [{{\displaystyle \mathrm{S}}}_2]}}_{\mathrm{t}}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_2}}}>>1$ and [S₁]_t, < [S₂]_t; then, [S₁] was chosen as the independent variable. Once more, the index t denotes that in the value of the ${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}$, and under these conditions, the total concentrations both of substrates should be considered.

[II]

Specific cases of esterifications following an ordered bi–bi mechanism:

[A]

With double dead-end substrate inhibition:

[i]

If the accepted conditions are that [S₁]_t ≈ [S₂]_t << K_mS1 and K_mS2, while $K_{\mathrm{i}\mathrm{A}}={\displaystyle \frac{{{\displaystyle k}}_{-1}}{{{\displaystyle k}}_1}}$, then the following is true:

$$\mathrm{v}={\displaystyle \frac{{[\mathrm{E}]}_{\mathrm{t}}{k}_{\mathrm{c}\mathrm{a}\mathrm{t}}[{\mathrm{S}}_1][{\mathrm{S}}_2]}{\frac{{{\displaystyle k}}_{-1}}{{{\displaystyle k}}_1}{K}_{\mathrm{m}{\mathrm{S}}_2}+K_{\mathrm{m}{\mathrm{S}}_1}[{\mathrm{S}}_2](1+\frac{[{{\displaystyle \mathrm{S}}}_2]}{{{\displaystyle K}}_{\mathrm{i}{\mathrm{S}}_2}})+K_{\mathrm{m}{\mathrm{S}}_2}[{\mathrm{S}}_1](1+\frac{{\displaystyle [{\mathrm{S}}_1]}}{{{\displaystyle K}}_{\mathrm{i}{\mathrm{S}}_1}})+[{\mathrm{S}}_1][{\mathrm{S}}_2]}}$$

According to the estimated values of the parameters in the above equation (Equation (A3), in Appendix A), as well as the proposed mechanistic details from the solvent isotope effect studies and the proton inventories, we can eliminate the term ${\displaystyle \frac{{{\displaystyle k}}_{-1}}{{{\displaystyle k}}_1}}{K}_{\mathrm{m}{\mathrm{S}}_2}$, as the value of k_-1 approaches zero ([4,5] and references therein). In that case, the above equation degenerates to Equation (A1) (in Appendix A), and therefore the relation ${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}=K_{\mathrm{m}{\mathrm{S}}_1}+K_{\mathrm{m}{\mathrm{S}}_2}$ is valid for the parameter ${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}$. The same holds true for all the relative rate equations of the ordered bi–bi mechanism, which are referenced herein. Similarly, and by considering the above conditions and assumptions, the ${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}$ values that should be assigned to the following cases are as stated below:

[ii]

If the accepted conditions are that [S₁]_t ≈ [S₂]_t >> K_mS1 and K_mS2, according to (i), the following is true:

$${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}={\displaystyle \frac{{{\displaystyle [\mathrm{S}]}}_{\mathrm{t}}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_1}}}(K_{\mathrm{m}{\mathrm{S}}_1}+K_{\mathrm{m}{\mathrm{S}}_2})$$

[iii]

If the accepted conditions are that [S₁]_t ≠ [S₂]_t >> K_mS1 and K_mS2, then, according to the assumptions, the following is true:

$${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}={\displaystyle \frac{{{\displaystyle K}}_{{{\displaystyle \mathrm{m}\mathrm{S}}}_1}{\displaystyle {{\displaystyle [{{\displaystyle \mathrm{S}}}_2]}}_{\mathrm{t}}^2}{{\displaystyle +K}}_{{{\displaystyle \mathrm{m}\mathrm{S}}}_2}{{\displaystyle [{\mathrm{S}}_1]}}_{\mathrm{t}}^2}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_1}{{\displaystyle [{{\displaystyle \mathrm{S}}}_2]}}_{\mathrm{t}}}}$$

A remark similar to the one for case [I]-[A]-[iii] should also be considered for this case.

[B]

With single dead-end substrate (alcohol) inhibition:

[i]

If the accepted conditions are that [S₁]_t ≈ [S₂]_t << K_mS1 and K_mS2, while [S₁] and [S₂] ≠ 0, then the following is true:

$${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}=K_{\mathrm{m}{\mathrm{S}}_1}+K_{\mathrm{m}{\mathrm{S}}_2}$$

[ii]

If the accepted conditions are that [S₁]_t ≈ [S₂]_t >> K_mS1 and K_mS2, while [S₁] and [S₂] ≠ 0, then the following is true:

$${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}={\displaystyle \frac{K_{\mathrm{m}{\mathrm{S}}_1}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_2}}}{\displaystyle {{\displaystyle [{\displaystyle \mathrm{S}}]}}_{\mathrm{t}}}+K_{\mathrm{m}{\mathrm{S}}_2}$$

[iii]

If the accepted conditions are that [S₁]_t ≠ [S₂]_t >> K_mS1 and K_mS2, while [S₁] and [S₂] ≠ 0, then the following is true:

$${{\displaystyle K}}_{\mathrm{m}}^{\mathrm{a}\mathrm{p}\mathrm{p}}={\displaystyle \frac{K_{\mathrm{m}{\mathrm{S}}_1}}{{{\displaystyle K}}_{\mathrm{i}{{\displaystyle \mathrm{S}}}_2}}}+K_{\mathrm{m}{\mathrm{S}}_2}{\displaystyle \frac{{{\displaystyle [{{\displaystyle \mathrm{S}}}_1]}}_{\mathrm{t}}}{{{\displaystyle [{{\displaystyle \mathrm{S}}}_2]}}_{\mathrm{t}}}}$$

A statement similar to Remark 2 should accompany this relation as well.

4. Conclusions

The techniques applied in this study were based on the use of reasonable approximations aimed at simplifying multiparametric and multivariable model equations. All achieved simplifications can be considered successful according to our present and previous results. Attempts to use multiparametric equations of two independent variables to fit experimental data fail without the necessary application of surface fitting techniques, which are difficult and usually generate erroneous results, depending on the algorithms used and appropriate programming. The proposed practice of modifying multiparametric and multivariable model equations, through the use of reasonable approximations and experimental and/or theoretical evidence, provides quick and errorless estimates of the values of significant kinetic parameters, useful in the productive areas of biotechnological and industrial applications. Within these areas, there is an increasingly focused demand for useful, rapid and significant information. We expect that the new know-how and feasible practice opened up in the context of this work will satisfy the demands and practical needs of a wide audience of scientists. The novelty of this study can easily be seen in these conclusions. In fact, this novelty is not based solely on the successfully applied approaches and the simplified rate equations that were subsequently revealed. The systematic methodology followed is the novelty that led to the formulation of these simplified equations, as well as the simplified formulas of the various Km parameters, depending on the case under consideration.