Automatic Calculation of the Kinetic Parameters of Enzymatic Reactions with Their Standard Errors Using Microsoft Excel

Abstract

We created a Microsoft Excel file, Enzyme_Kinetics_Calculator, which includes macro programs that automatically calculates kinetic parameters for typical kinetic equations of enzymatic reactions, accompanied by their standard errors, by minimizing the residual sum of squares thereof. The [S]-v plot is automatically drawn with the theoretical lines and, similarly, the 1/[S]-1/v plot in the case of linear theoretical lines. Enzyme_Kinetics_Calculator is available as a supplementary file for this paper (see J. Appl. Glycosci. Web site).

Abbreviations

GRG, generalized reduced gradient method; RSS, residual sum of squares.

The least squares method is generally used to calculate parameters of a theoretical formula from observed data, which minimizes the residual sum of squares (RSS). The relationship between the enzymatic reaction rate and the concentrations of the substance/s (substrate/s and inhibitor) is given by the Michaelis-Menten equation (1) and its derivatives^1)2)3)4)5) with several kinetic parameters, such as K_m and V_max. Unfortunately, the least squares method cannot be directly applied to such equations for enzyme kinetics because the parameters in the equations cannot be separated from each other.⁶⁾ Therefore, curve fitting is used to calculate enzyme kinetics.

  

$$v=\frac{V_{\max }\left[S\right]}{K_{\mathrm{m}}+\left[S\right]}$$(1)

Curve fitting can be performed using commercially available graphing software, which is generally purchased. A method for calculating parameters using R, an open-source programming language for statistical analysis, has been reported.⁷⁾ However, the use of R language requires some knowledge of computer programming

Microsoft Excel (Microsoft Corporation, Redmond, WA, USA) is one of the most popular spreadsheet software packages. Curve fitting can be performed to minimize the RSS by the generalized reduced gradient method (GRG) using the solver function in Microsoft Excel, which is a standard add-in. However, the solver does not calculate the standard errors of the optimized parameters.

Therefore, we created an Excel file, Enzyme_Kinetics_Calculator, including macro programs that automatically calculates kinetic parameters from the data of enzyme reaction rates at various concentrations of substrate/s (and an inhibitor). It is available in Excel 2016 and later versions. We prepared four equations for single-substrate kinetics (Michaelis-Menten, substrate inhibition, sigmoid where the Hill's number is a variant, and sigmoid where the Hill's number is a constant), four for inhibition kinetics (mix-type, competitive, uncompetitive, and non-competitive), and three for two-substrate kinetics (sequential bi bi, ping pong bi bi, and competitive substrate inhibition⁵⁾), as shown in Table 1. The calculated parameters are displayed along with their standard errors on the spreadsheet. The [S]-v plot is automatically drawn with theoretical lines and, similarly, the 1/[S]-1/v plot in the case of linear theoretical lines. Enzyme_Kinetics_Calculator is available as a supplementary file for this paper (see J. Appl. Glycosci. Web site).

Table 1. Kinetic equations prepared in Enzyme_Kinetics_Calculator.

Pattern	Kinetic equation
Michaelis-Menten	$$v=\frac{V_{\max }\left[S\right]}{K_{\mathrm{m}}+\left[S\right]}$$
Substrate-inhibition	$$v=\frac{V_{\max }}{1+\frac{K_{\mathrm{m}}}{\left[S\right]}+\frac{\left[S\right]}{K_{\mathrm{i}}}}$$
Sigmoid (n is a variant)	$$v=\frac{V_{\max }{\left[S\right]}^n}{K_{\mathrm{m}}^n+{\left[S\right]}^n}$$
Sigmoid (n is a constant)	$$v=\frac{V_{\max }{\left[S\right]}^n}{K_{\mathrm{m}}^n+{\left[S\right]}^n}$$
Mix-type inhibition	$$v=\frac{V_{\max }\left[S\right]}{K_{\mathrm{m}}\left(1+\frac{\left[I\right]}{K_{\mathrm{iC}}}\right)+\left[S\right]\left(1+\frac{\left[I\right]}{K_{\mathrm{iU}}}\right)}$$
Competitive inhibition	$$v=\frac{V_{\max }\left[S\right]}{K_{\mathrm{m}}\left(1+\frac{\left[I\right]}{K_{\mathrm{iC}}}\right)+\left[S\right]}$$
Uncompetitive inhibition	$$v=\frac{V_{\max }\left[S\right]}{K_{\mathrm{m}}+\left[S\right]\left(1+\frac{\left[I\right]}{K_{\mathrm{iU}}}\right)}$$
Noncompetitive inhibition	$$v=\frac{V_{\max }\left[S\right]}{(K_{\mathrm{m}}+\left[S\right])\left(1+\frac{\left[I\right]}{K_{\mathrm{i}}}\right)}$$
Sequential bi bi	$$v=\frac{V_{\max }\left[A\right]\left[B\right]}{K_{\mathrm{iA}}{K}_{\mathrm{mB}}+K_{\mathrm{mA}}\left[B\right]+K_{\mathrm{mB}}\left[A\right]+\left[A\right]\left[B\right]}$$
Ping-pong bi bi	$$v=\frac{V_{\max }\left[A\right]\left[B\right]}{K_{\mathrm{mA}}\left[B\right]+K_{\mathrm{mB}}\left[A\right]+\left[A\right]\left[B\right]}$$
Competitive substrate inhibition	$$v=\frac{V_{\max }\left[A\right]\left[B\right]}{K_{\mathrm{iA}}{K}_{\mathrm{mB}}+\left(K_{\mathrm{mA}}+\frac{K_{\mathrm{iA}}{K}_{\mathrm{mB}}}{K_{\mathrm{I}1}}\right)\left[B\right]+K_{\mathrm{mA}}\left[A\right]+\left[A\right]\left[B\right]+\left(\frac{K_{\mathrm{iA}}{K}_{\mathrm{mB}}}{K_{\mathrm{I}1}{K}_{\mathrm{I}2}}+\frac{K_{\mathrm{mA}}}{K_{\mathrm{I}1}}\right){\left[B\right]}^2+\frac{K_{\mathrm{mA}}}{K_{\mathrm{I}1}{K}_{\mathrm{I}2}}{\left[B\right]}^3}$$

In the macro programs, the RSS is calculated from the observed data with the initial estimations of the parameters from the corresponding theoretical equation under the assumption that the errors are evenly included in the velocity data. The optimal parameter values are calculated by changing the parameters until convergence to minimize the RSS using the solver function with the GRG. The standard error of each parameter is calculated automatically, as described below.

Consider the data set of (v, S), where v is the velocity of the reaction and S is the concentration/s of the substance/s involved in the enzymatic reaction, such as [S] for single-substrate kinetics, ([S], [I]) for inhibition kinetics, and ([A], [B]) for two-substrate kinetics. The kinetic equation is described as a function of concentration/s and parameters (2). P represents the set of parameters (such as K_m and V_max, in the Michaelis-Menten equation). “p_j” represents the j^th parameter, and “m” represents the number of the parameters in the equation.   

$$v=f\left(P\mathrm{:}S\right)\left(P=\left(\begin{array}{c}{p}_1\\ ⋮\\ p_m\end{array}\right)\right)$$(2)

From the n sets of (v_i, S_i), the RSS and standard error of v (δ_v) are calculated as in equations (3) and (4).   

$$\text{RSS}=\sum _{i=1}^n{\{v_i-f(P\mathrm{:}{S}_i)\}}^2$$(3)

  

$${\mathrm{\delta }}_v=\sqrt{\frac{\mathit{RSS}}{n-m}}$$(4)

Equation (2) is transformed into equation (5) by considering the first terms of the Taylor expansion for each parameter. Pe (set of constants) is the initial estimation of P (set of variants).   

$$v=f\left(Pe\mathrm{:}S\right)+\sum _{j=1}^m\left\{\left(p_j-p_{j}e\right)\frac{\partial f\left(Pe\mathrm{:}S\right)}{\partial p_j}\right\}\left(Pe=\left(\begin{array}{c}{p}_{1}e\\ ⋮\\ p_{m}e\end{array}\right)\right)$$(5)

Because equation (5) is a first-order equation of the parameter set, P, the least squares method can be applied directly. When substituting the optimized parameter values for Pe, P is calculated to be equal to Pe using the least squares method. The RSS is calculated from (5), as in equation (6).   

$$\text{RSS}={\sum _{i=1}^n\left[v_i-f\left(Pe\mathrm{:}{S}_i\right)-\sum _{j=1}^m\left\{\left(p_j-p_{j}e\right)\frac{\partial f\left(Pe\mathrm{:}{S}_i\right)}{\partial p_j}\right\}\right]}^2$$(6)

To minimize the RSS, the following simultaneous equations with m unknowns (7) should be solved.   

$$\frac{\partial \mathit{RSS}}{\partial p_j}=0\left(1\le j\le m\right)$$(7)

Consider a square matrix C of size m, where its element at the k^th row and l^th column, c_k,l, is as shown in equation (8).   

$$c_{k,l}=\sum _{i=1}^n\left(\frac{\partial f\left(Pe\mathrm{:}{S}_i\right)}{\partial p_k}·\frac{\partial f\left(Pe\mathrm{:}{S}_i\right)}{\partial p_l}\right)$$(8)

The parameters are calculated from equation (9).   

$$\begin{array}{c}P=Pe+C^{-1}\left(\begin{array}{c}{d}_1\\ ⋮\\ d_m\end{array}\right)\\ \left(d_j=\sum _{i=1}^n\left[\left\{v_i-f\left(Pe\mathrm{:}{S}_i\right)\right\}\frac{\partial f\left(Pe\mathrm{:}{S}_i\right)}{\partial p_j}\right]\right)\end{array}$$(9)

The standard deviation of each parameter is calculated using equation (10) based on the law of propagation of errors, where M_j,j is the minor determinant of matrix C with the j^th row and column.   

$${\mathrm{\delta }}_{P_j}={\mathrm{\delta }}_v\sqrt{\frac{M_{j,j}}{\mathit{det}\left(C\right)}}\left(1\le j\le m\right)$$(10)

We now explain how to use the Enzyme_Kinetics_Calculator. Immediately after opening the Excel file, the users are requested to enable the macros in the file. If the macros do not work, users are prompted to load the solver add-in, as instructed at the URL (https://support.microsoft.com/en-us/office/load-the-solver-add-in-in-excel-612926fc-d53b-46b4-872c-e24772f078ca, retrieved Nov. 9, 2022). They can then move to the sheet for the desired rate equation from the front page by clicking the corresponding button. They can return to the front page from each sheet by clicking the “return to the front page” button if necessary.

Here, we explain the calculation using the Michaelis-Menten kinetic sheet (Fig. 1). Only the colored cells are editable, and the sample data are displayed initially. After erasing the sample data, users can enter the reaction rates accompanied by the concentrations of the substrate in the yellow cells and the initial estimates of K_m and V_max in the green cells. After input, the [S]-v and 1/[S]1/v plots are automatically displayed. The solid lines in the graphs represent the lines prepared using the initial estimations of the parameters. Then, the parameters are optimized using the solver function by clicking the “calculate” button. The graphs on the sheet are editable. Although it is necessary to determine the standard error of V_max/K_m, which is an index of catalytic efficiency, to evaluate the validity of the data, it cannot be calculated from the standard errors of K_m and V_max. The error of V_max/K_m is calculated based on the modified Michaelis-Menten equation expressed as Ce (= V_max/K_m) and K_m in equation (11). It should be noted that the standard error of K_m is calculated to be the same regardless of the equation used.

  

$$v=\frac{Ce\left[S\right]}{1+\frac{\left[S\right]}{K_{\mathrm{m}}}}$$(11)

Fig. 1. The sheet for the Michaelis-Menten kinetic.

Next, we explain the mix-type inhibition sheet as an example (Fig. 2). After erasing the sample data, the reaction rates at various concentrations (up to 10) of the substrate with various concentrations of the inhibitor (up to 10, including the absence of the inhibitor) can be entered into the yellow cells. If there are cells in the matrix that are not measured, the data will be ignored if the corresponding cells remain blank. After entering the initial estimations of the parameters in the green cells, the user should click the “calculate” button to optimize the parameters with their standard errors. The [S]-v plot and the 1/[S]-1/v plot at various [I] values are automatically drawn.

Fig. 2. The sheet for the mix-type inhibition kinetic.

At last, we demonstrate an analysis of the historical data. A set of data on an invertase reaction (hydrolysis of sucrose detected as the change in the optical rotation) reported by Michaelis and Menten¹⁾²⁾³⁾ (adopted from the “Results of the experiment in Table II” in the supplement information of the reference³⁾) was analyzed using Enzyme_Kinetics_Calculator. We selected substrate-inhibition kinetic sheet because of a small decrease in the reaction rate at higher substrate concentrations, even though Michaelis and Menten did not state about substrate inhibition.¹⁾²⁾³⁾ Starting from initial estimations of the parameters (K_m, K_i, V_max) as (0.02 mol/L, 2 mol/L, 0.8 °/min), they were regressed to be (0.023 ± 0.003 mol/L, 2.25 ± 0.63 mol/L, 0.088 ± 0.004 °/min) to give a theoretical curve corresponding well with the experimental data as shown in Fig. 3.

Fig. 3. Analysis of the historical data on the enzymatic hydrolysis of sucrose reported by Michaelis and Menten using Enzyme_Kinetics_Calculator.

CONFLICTS OF INTEREST

The authors declare no conflicts of interest associated with this manuscript.

ACKNOWLEDGMENTS

This work was supported by JSPS KAKENHI Grant Number 21K19070. We thank Professor Shinya Fushinobu of the University of Tokyo for investigating the usability and bugs of the program. Thanks are also due to Editage (www.editage.com) for English language editing.

REFERENCES

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