A Recurring Misconception Regarding the Fitting and Plotting of Enzyme Kinetics Data Leads to the Loss of Significant Reaction Parameters and Rate Constants

Abstract

In recent years, an increasing number of published articles on biocatalysis have reported new enzymes that exhibit biotechnologically interesting catalytic activities. Some authors rush to publish their work in an attempt to ensure “novelty” and often present their results with various types of graphs, such as scatter plots or line-and-symbol plots using nonexistent parameters as ordinates, e.g., “Relative Activity (%)” and the like, vs. abscissae such as pH value, temperature, etc. Nevertheless, in vitro enzyme kinetics remains a valuable tool for understanding, optimizing, and effectively applying these biocatalysts under diverse reaction conditions, taking advantage of all available experimental results. This work aims to emphasize the importance of processing experimental data from enzymatic reactions through appropriate fitting with known kinetic equations and to discourage the use of nonexistent parameters. To support this integrated approach, specific examples are provided for the calculation of significant kinetic and thermodynamic parameters, as well as rate constants.

1. Introduction

The demand for new enzymes is rapidly increasing in the fields of agro-industrial applications, GRAS food processing, pharmaceuticals and cosmetics, soil sciences, pest control, and other activities of biotechnological interest [1,2]. These enzymes are mainly produced through the bioconversion of agro-industrial and similar types of waste from cereal crops, as well as from other sources using simple, cost-effective, and low-energy bioprocesses [3,4,5]. In general, the catalytic efficiency and specificity of enzymes depend on their structures (primary, secondary, tertiary, etc.) and consequently on the structural features of their active and catalytic sites. Any structural reorganization is reflected in the experimentally measured values of the physically existent parameters, which are also affected by various factors of the reaction medium. The most common factors include pH value, absolute temperature, and the ionic strength of the reaction medium, among others [6,7,8,9]. It is widely recognized among enzymologists that the characterization, effectiveness, and application of any enzyme as a biocatalyst are evaluated and specified by considering the experimentally estimated values of its in vitro kinetic and thermodynamic parameters, crucial rate constants, and other relevant physically existent parameters. The most common factors include pH value, absolute temperature, and ionic strength of the reaction medium, among others [6,7,8,9]. It is widely recognized among enzymologists that the characterization, effectiveness, and application of any enzyme as a biocatalyst are evaluated and specified by considering the experimentally estimated values of its in vitro kinetic and thermodynamic parameters, crucial rate constants, and other relevant physically existent parameters.

Nevertheless, the enzyme kinetics tool should be properly applied, as it becomes ineffective when nonexistent parameters are involved. Important information that could be extracted from the available experimental data will also be lost [10]. Therefore, nonexistent parameters (e.g., relative activity %) cannot provide information about any other parameter, rate constant, etc., whose value could affect the catalytic effectiveness of the enzyme under study. Furthermore, the use of nonexistent parameters is misleading and dangerously inaccurate, especially when new enzymes are implemented in biocatalytic biotechnological processes, such as environmental pollution control, pesticide treatment and/or removal, or pharmaceutical products. In such cases, a knowledge of the values of the significant parameters and other relevant factors (e.g., the activation thermodynamic parameters ΔH^‡, ΔG^‡, and ΔS^‡) is crucial for the successful application of this enzyme, considering its cost, catalytic effectiveness, and the resulting outcomes [7,8,11,12]. In the simplest case where the enzyme under consideration follows Michaelis–Menten kinetics, the relevant physically existent parameters are k_cat/K_m, k_cat, and K_m, as well as the rate constants for both the reversible and irreversible steps, among other important factors that will be discussed below.

The Michaelis–Menten parameters, k_cat/K_m, k_cat, and K_m, are usually calculated by processing and recording repeated measurements under conditions where either the relationships [S] << K_m or [S] >> K_m (saturation of the enzyme by the substrate) apply. When substrate saturation is not possible, initial reaction rates should be measured across a broad range of substrate concentrations under well-defined conditions. The resulting experimental data should then be fitted by the Michaelis–Menten equation for proper analysis. Therefore, only fitting and plotting the Michaelis–Menten parameters vs. pH value, absolute temperature, and/or other relevant factors can provide meaningful and reliable results, while using nonexistent parameters for such analyses is inappropriate and should be strictly avoided.

2. Results and Discussion

2.1. Explanatory Examples

Figure 1 illustrates the selected representative examples of fitting non-polynomial, non-linear equations to experimental data derived from real enzyme kinetic measurements. In these cases, by using appropriate fitting procedures, significant information about the catalytic capacity of the enzyme under investigation was obtained. These include the estimated values of the kinetic parameters (k_cat/K_m, k_cat, and K_m), the rate constants (k₁, k₋₁, k_2, etc.) and their activation energies, the activation thermodynamic parameters (ΔH^‡, ΔG^‡, and ΔS^‡), as well as other parameters, such as pK_a, etc.

The obvious differences in the pH profiles of the plots (a) in Figure 1, both in terms of shape (narrower profile for k_cat/K_m and wider for k_cat) and in terms of the pH values where the maximum estimates of k_cat/K_m and k_cat were observed, can be easily interpreted through Equations (1)–(3) and Scheme 1a. In addition, it has been pointed out that the pH dependences of k_cat/K_m and k_cat refer to different steps of the three-step enzymatic reaction model (Scheme 1a). Furthermore, the estimated pK_a values contribute to elucidating the role of catalytic sites, either as stronger or weaker nucleophiles (e.g., the catalytic His, in the case of serine proteases), providing information on the possible formation or breakdown of hydrogen bonds with other residues of the catalytic site. Pronounced differences are also observed in the absolute temperature profiles of the parameters k_cat/K_m and k_cat. The plots of Figure 1b differ both in shape and in the absolute temperatures where the maximum estimates of k_cat/K_m and k_cat were obtained. These absolute temperature profiles provide important information regarding the estimated values of the rate constants k₁, k_-1, and k₂ (Scheme 1a) through Equations (8) and (9) and reveal the rate-limiting step of the overall reaction, according to Equation (1), depending on whether relation k₋₁ > k₂ or k₋₁ < k₂ holds. Furthermore, the values of the activation thermodynamic parameters ΔH^‡ and ΔS^‡ can be easily estimated through Equation (10) by plotting, as a response variable, the expression T_i × [ln(k_i/T_i)] vs. the abscissa T_i. Thereafter, from the intercept and slope of the straight-line graph, the values of the activation thermodynamic parameters ΔH^‡ and ΔS^‡ can be easily calculated. The activation thermodynamic parameter ΔG^‡ can be calculated through the relation ΔG^‡ = ΔH^‡ − TΔS^‡, which can only be used under constant pressure and temperature, i.e., conditions prevailing in a biochemical laboratory [6,9,11,14,15,16,17]. It should be noted that the correct choice of the reference temperature T₀ is a prerequisite for the valid use of the Eyring equation.

The profiles of the Michaelis–Menten parameters, in Figure 1, both in terms of pH value and absolute temperature, strongly support the aim and novelty of this work. Furthermore, satisfactory experimental evidence has been provided to any potential experimenter regarding the futile and completely pointless use of nonexistent parameters. In addition, the profiles in Figure 1 clearly showed that through the proposed methodology, significant information can be obtained regarding the functional properties and catalytic efficiency of the enzyme used, whether this is expressed through a three-step or a multi-step reaction model (Scheme 1a,b) [6,7,9,10,11,12,18]. Therefore, clear evidence has been provided that the values of all the aforementioned parameters and rate constants are not only necessary to be calculated but also constitute important indicators of the functioning of all enzymes, as well as their overall catalytic potential and activity.

2.2. Validation of the Explanatory Examples

Only a few examples are presented in Figure 1. However, more examples can be found in the literature, which are informative and significant, giving a clear picture of the very distinct differences in the behavior of enzymes during catalysis under different conditions (either [S] << K_m or [S] >> K_m), even in the same reaction medium. This latter is true for the cases of all profiles depicted in Figure 1, and researchers in the field of enzymology and/or enzyme biotechnology should take it seriously.

According to a recurring misconception and the subsequent use of nonexistent parameters, e.g., “Related Activity (%)” as an ordinate, it has been proven unambiguously that all the aforementioned significant information is lost. Consequently, important properties of the enzyme under study may remain unknown, greatly negatively affecting any biotechnological process in which this biocatalyst could be applied. Let us consider the consequences and costs of the loss of scientific knowledge, as well as the economic revenue of any creative, academic, and/or industrial production.

Unfortunately, some authors of a considerable number of recent manuscripts, both submitted and published, continue to misapprehend scientifically established guidelines regarding the fitting and plotting of experimental enzyme kinetic data. These authors may be unaware that such practices are incorrect and thus miss values of significant existing kinetic and thermodynamic parameters, as well as crucial rate constants, whereas they could easily extract important information from their experimental data.

3. Methods and Tools

3.1. Summary

To facilitate the discussion regarding the aims of this work, the Michaelis–Menten Equation (1), corresponding to Scheme 1a, was chosen as an example. In cases where the relations either [S] << K_m or [S] >> K_m hold, then the Michaelis–Menten equation degenerates into Equations (2) or (3), respectively.

$$\mathrm{v}={\displaystyle \frac{V_{\max }[\mathrm{S}]}{\frac{{{\displaystyle k}}_{-1}+{{\displaystyle k}}_2}{{{\displaystyle k}}_1}+[\mathrm{S}]}}={\displaystyle \frac{V_{\max }[\mathrm{S}]}{K_{\mathrm{m}}+[\mathrm{S}]}}={\displaystyle \frac{k_{\mathrm{cat}}{[\mathrm{E}]}_0[\mathrm{S}]}{K_{\mathrm{m}}+[\mathrm{S}]}}={\displaystyle \frac{\frac{{{\displaystyle k}}_{\mathrm{cat}}}{{{\displaystyle K}}_{\mathrm{m}}}{[\mathrm{E}]}_0[\mathrm{S}]}{1+\frac{{\displaystyle [\mathrm{S}]}}{{{\displaystyle K}}_{\mathrm{m}}}}}={\displaystyle \frac{\frac{{{\displaystyle k}}_{\mathrm{cat}}}{{{\displaystyle K}}_{\mathrm{m}}}{[\mathrm{E}]}_0}{\frac{1}{{\displaystyle [\mathrm{S}]}}+\frac{1}{{{\displaystyle K}}_{\mathrm{m}}}}}$$

(1)

$$\mathrm{v}={\displaystyle \frac{V_{\max }[\mathrm{S}]}{K_{\mathrm{m}}+[\mathrm{S}]}}={\displaystyle \frac{k_{\mathrm{cat}}{[\mathrm{E}]}_0[\mathrm{S}]}{K_{\mathrm{m}}+[\mathrm{S}]}}\approx {\displaystyle \frac{k_{\mathrm{cat}}}{K_{\mathrm{m}}}}{[\mathrm{E}]}_0[\mathrm{S}]$$

(2)

$$\mathrm{v}={\displaystyle \frac{V_{\max }[\mathrm{S}]}{K_{\mathrm{m}}+[\mathrm{S}]}}={\displaystyle \frac{k_{\mathrm{cat}}{[\mathrm{E}]}_0[\mathrm{S}]}{K_{\mathrm{m}}+[\mathrm{S}]}}\approx {\displaystyle \frac{k_{\mathrm{cat}}{[\mathrm{E}]}_0[\mathrm{S}]}{[\mathrm{S}]}}=k_{\mathrm{cat}}{[\mathrm{E}]}_0$$

(3)

k_cat/K_m (or V_max/K_m) and k_cat (or V_max) are the second- and first-order rate constants, according to Equations (2) and (3), respectively, and refer to the simplest three-step enzymatic reaction model of Scheme 1a. The units of kcat/Km and kcat are M⁻¹ s⁻¹ and s⁻¹, respectively. The k_cat/K_m is related to the short, quasi-straight-line segment of the Michaelis–Menten hyperbola, which coincides with its tangent near the intersection of the ordinate and abscissa axes, according to Equation (2) and the condition [S] << K_m.

However, k_cat is related to another short, quasi-straight-line, which coincides with the tangent near the saturation region of the enzyme to its substrate, according to Equation (3) and the condition [S] >> K_m. Both parameters k_cat/K_m and k_cat are existent parameters (Scheme 1a and/or Scheme 1b). There is no other quasi-straight-line segment in the Michaelis–Menten hyperbola, except for the two mentioned above. Therefore, no other existent parameter can be defined, except for the two parameters k_cat/K_m and k_cat [6,7,8,11]. The parameter K_m (expressed in units M = mol/L) is an important existing parameter, although its physical meaning depends on the experimental conditions of the particular enzymatic reaction under consideration.

3.2. Useful Remarks

The physical meaning of the Michaelis–Menten parameters, according to Equations (1)–(3), as well as to Scheme 1a, is the key for the researcher to obtain additional and in-depth information about crucial properties and potential uses of any enzyme under consideration [6,7,8,12,15]. Accordingly, the

• k_cat/K_m = (k₁ × k₂)/(k₋₁ + k₂) refers to the reaction of the free enzyme with the free substrate, towards the formation of the product(s) and the release of the enzyme, in order to be ready for the next catalytic cycle, in cases where Equation (2) holds. Consequently, the profiles of k_cat/K_m vs. either the pH value or absolute temperature or other factors reveal changes in the free enzyme and free substrate molecules.
• k_cat = k₂ refers to the irreversible step of the three-step enzymatic reaction model (Scheme 1a), according to Equation (3), and therefore the profiles of kcat vs. the abovementioned factors reveal changes in the enzyme–substrate complex (ES).
• K_m = (k₋₁ + k₂)/k₁ generally represents the substrate concentration that drives the enzymatic reaction to half its maximum value and could be characterized as the dissociation constant of the ES complex only under specific conditions (Scheme 1a). Km profiles vs. the aforementioned factors reveal changes in enzyme–substrate complexes (ES, ES′, etc.); however, such profiles are beyond the aims of this work and will not be discussed further.

3.3. Subject Matter

Specific equations were developed and used to calculate the estimates of the significant kinetic parameters, activation thermodynamic parameters, and rate constants, adding value to any relevant study submitted for publication [8,9,11,12,18]. The dependence of the Michaelis–Menten parameters vs. pH value is generally expressed by Equation (4), which can be simplified to either Equation (5) or (6), and/or to another more complex one. The cases where two pK_a values are calculated, one in each of the two limbs of the pH profile curve, the acidic and the basic, are described by Equation (5), whereas the cases where three pK_a values are calculated, one in the acidic limb and two in the basic limb of the pH profile curve, are described by Equation (6). In Equations (4)–(6), (k)_obs, (k)^lim, n, and B_ij are the response variable (i.e., the experimentally observed values of the parameter and/or rate constant under study), its limiting value for the hydronic state XH_i−1, the number of active hydronic states, and a specific description referred to as two particular matrices (Table S1 in the supplement), respectively; in Equations (5) and (6), the independent variable is the pH value. However, in cases where there are two or more pK_a values per limb of the pH profile curve under study, the corresponding equations can be appropriately derived from Equation (4) [8,14,19,20].

$${\left(k\right)}_{\mathrm{obs}}={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}\frac{{\left(k\right)}_{{\mathrm{XH}}_{\mathrm{i}-1}}^{\lim }}{\left(1+{\displaystyle \sum _{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{B}}_{\mathrm{i},\mathrm{j}}}\right)}}$$

(4)

$${\left(k\right)}_{\mathrm{obs}}={\displaystyle \frac{{\left(k\right)}^{\lim }}{1+{10}^{\mathrm{p}{K}_{\mathrm{a}1}-\mathrm{pH}}+{10}^{\mathrm{pH}-\mathrm{p}{K}_{\mathrm{a}2}}}}$$

(5)

$${\left(k\right)}_{\mathrm{obs}}={\displaystyle \frac{{\left(k\right)}^{\lim }}{1+10^{\mathrm{p}{K}_{\mathrm{a}1}+\mathrm{p}{K}_{\mathrm{a}2}-2\mathrm{pH}}+{10}^{\mathrm{p}{K}_{\mathrm{a}2}-\mathrm{pH}}+{10}^{\mathrm{pH}-\mathrm{p}{K}_{\mathrm{a}3}}}}$$

(6)

Furthermore, using the Arrhenius Equation (7), Equations (8) and (9), which relate the dependences of the Michaelis–Menten parameters vs. the absolute temperature, were developed. In Equation (7), (k)_obs, (k)^lim, E_k, R, T_i, and T₀ are the response variables (i.e., the experimentally measured values of the parameter under study at the absolute temperature T_i), its limiting value at the reference temperature T₀, the activation energy corresponding to the parameter under study (i.e., k_cat/K_m, k_cat, k₁, k₋₁, k₂, etc.), the gas constant (8.314 J × mol⁻¹ × K⁻¹), the absolute temperature where the experimental values of the studied parameters were measured, and the reference temperature, respectively. Equation (10) is a suitable linear form of the Eyring equation [8,9,11,14,16]. Therefore, significant rate constants (k₁, k₋₁, k₂, etc.), the activation thermodynamic parameters, ΔH^‡, ΔG^‡, and ΔS^‡, as well as activation energies corresponding to the mentioned rate constants, can be calculated from the absolute temperature profiles.

$${(k)}_{\mathrm{obs}}={(k)}^{\lim } {{\displaystyle \mathrm{e}}}^{\left[-{\displaystyle \frac{{\mathrm{E}}_k}{\mathrm{R}}}\left({\displaystyle \frac{1}{{\mathrm{T}}_{\mathrm{i}}}} - {\displaystyle \frac{1}{{\mathrm{T}}_0}}\right)\right]}$$

(7)

$${\left(\frac{k_{\mathrm{cat}}}{K_{\mathrm{m}}}\right)}_{\mathrm{obs}}={\displaystyle \frac{{\alpha }_0{\mathrm{e}}^{\left(\frac{{\mathrm{E}}_{\alpha }}{\mathrm{R}}\left(\frac{1}{{\mathrm{T}}_{\mathrm{i}}}-\frac{1}{{\mathrm{T}}_0}\right)\right)} {\left(k_1\right)}_0{\mathrm{e}}^{\left(-\frac{{\mathrm{E}}_1}{\mathrm{R}} \left(\frac{1}{{\mathrm{T}}_{\mathrm{i}}}-\frac{1}{{\mathrm{T}}_0}\right)\right)}}{1+{\alpha }_0{\mathrm{e}}^{\left(\frac{{\mathrm{E}}_{\alpha }}{\mathrm{R}} \left(\frac{1}{{\mathrm{T}}_{\mathrm{i}}}-\frac{1}{{\mathrm{T}}_0}\right)\right)}}}$$

(8)

$${\left(k_{\mathrm{cat}}\right)}_{\mathrm{obs}}={\left(k_{\mathrm{cat}}\right)}^{\lim } {\mathrm{e}}^{\left[-{\displaystyle \frac{{\mathrm{E}}_{k_{\mathrm{cat},0}}}{\mathrm{R}}}\left({\displaystyle \frac{1}{{\mathrm{T}}_{\mathrm{i}}}}-{\displaystyle \frac{1}{{\mathrm{T}}_0}}\right)\right]}$$

(9)

T_i × [ln(k_i/T_i)] = T_i × [ln(k_B/ħ) + ΔS^‡/R] − ΔH^‡/R

(10)

In Equations (8) and (9), a₀, k₁, E₁, and E_a are the ratio k₂/k₋₁, the rate constant k₁ (k₁, k₋₁, and k₂ appeared in Scheme 1a), the activation energy that corresponds to the rate constant k₁, and the difference E₋₁ − E₂, respectively; E₋₁ and E₂ are the activation energies corresponding to the rate constants k₋₁ and k₂, respectively. The independent variable of Equations (8) and (9) is the expression $\left(\frac{1}{{\mathrm{T}}_{\mathrm{i}}} - \frac{1}{{\mathrm{T}}_0}\right)$.

In Equation (10), k_i refers to the experimentally measured values of the parameter (k_cat/K_m or k_cat) or the rate constants (k₁, k₋₁, k_2, etc.) under study at absolute temperature T_i, whereas T_i, k_B, and ħ, are the absolute temperatures at which the experimental values of the parameters under study were measured, the Boltzmann’s constant (1.381 × 10⁻²³ J × K⁻¹), and the Planck’s constant (6.626 × 10⁻³⁴ J × s), respectively. The expression T_i × [ln(k_i/T_i)] and the absolute temperature T_i are the response variable and the independent variable, respectively, in Equation (10), whereas the expression ln(k_B/ħ) has the dimensionless constant arithmetic value of 23.76.

4. Conclusions

This work is primarily addressed toward researchers focused on the characterization and application of enzymes for biotechnological and related purposes. It is essential for these authors to recognize the importance of seeking out and uncovering “hidden”, yet significant, scientific information within their experimental data. Throughout this study, convincing arguments have been presented regarding the significance, objectivity, and utility of reporting results based on physically meaningful parameters. There are at least two key reasons supporting this rigorous approach. First, the accurate determination and reporting of significant physically existent parameters, namely the Michaelis–Menten, rate constants (k₁, k₋₁, k₂ and k₃), activation thermodynamic parameters (ΔH^‡, ΔG^‡, and ΔS^‡), and other relevant parameters, such as pK_a values, are fundamental for understanding and utilizing the catalytic potential of new enzymes. Conclusions should therefore be grounded in the calculated values of these parameters to fully appreciate and utilize the enzymes’ capabilities. Second, the use of nonexistent parameters is both inappropriate and ineffective, leading to the unnecessary expenditure of time and resources without yielding meaningful scientific knowledge. As discussed, experimenters are expected to benefit significantly by focusing on the observed dependences of these established parameters in response to variations in the pH value, absolute temperature, and other relevant factors. In summary, researchers are strongly advised to adopt the experimental methodologies outlined in this work, as doing so will enhance the reliability and impact of their findings, ultimately advancing the field of enzyme biotechnology.