Development of a mathematical model for microbial consumption of glucose and fructose during cocoa (Theobroma cacao L.) fermentation process

Abstract

A fundamental procedure for achieving the highest quality of cocoa is the fermentation of the bean carried out by microorganisms, a process in which the flavor and aroma of chocolate are developed. The objective of this research was to develop a mathematical model of microbial sugar consumption during the cocoa fermentation process. In this way, it is possible to obtain a mathematical tool that allows the representation of sugar consumption during the fermentation process. Initially, the need to construct a mathematical model was proposed due to its economic importance for small-scale producers in the national and international industries. In addition to the lack of control over the fermentation process, the complexity in terms of the microbial communities involved yeast (Y), lactic acid bacteria (LAB), and acetic acid bacteria (AAB). The methodological approach included (i) bibliographic review; (ii) specification of the parameters of the model, state, initial values, and equations; (iii) implementation of the mathematical model with the parameters, state, and initial values defined for each investigation; and (iv) validation of the model by comparing it with the experimental data from each publication. As a result, the model presented here for microbial sugar consumption during cocoa fermentation has the ability to represent each of the selected data points with significantly high precision. Furthermore, the proposed model aims to enhance the quality, efficiency, and sustainability of cocoa production, as well as contribute to scientific research related to this process. However, it is necessarily a simplification of the various biological processes that occur during cocoa bean fermentation.IMPORTANCECreating a mathematical model based on ordinary differential equations for glucose and fructose consumption by Y and LAB during cocoa fermentation is vital for advancing scientific and industrial practices. Cocoa fermentation is a complex, multi-step process involving microbial communities that play a key role in developing the flavor and aroma of chocolate. A robust mathematical model helps to understand the interactions between these microorganisms and sugar dynamics, providing a clearer understanding of the fermentation process.

Keywords: cocoa; fermentation; microorganisms; model; simulation.

Fig 1

Graphical representation of the model against the experimental data. (a) Glucose (23), (b) Glucose (21), (c) Fructose (24), and (d) Fructose (25). Panels (a) and (b) show model simulations for glucose consumption, with experimental data from (23) and (21), respectively. Panels (c) and (d) display the corresponding simulations for fructose consumption using datasets from (24) and (25). The high coefficients of determination (R² values of 0.897, 0.948, 0.9231, and 0.9121) along with low RMSE values demonstrate that the model accurately captures the dynamics of sugar depletion. Moreover, the Kolmogorov-Smirnov tests yielded test statistics of D = 0.4286, 0.2941, 0.2, and 0.3077 with P-values exceeding 0.05, supporting the null hypothesis that there is no significant difference between the experimental data and the model predictions. These results confirm that the underlying assumptions and parameter estimations—based on experimental conditions and microbial growth kinetics—are robust and align well with observed fermentation behavior.

Fig 2

Regression Analysis of the ODE Numerical Solution. Adjustment of the one numerical solution of the ODE to different regression models. (a) Linear regression model: ${\displaystyle y\left(x\right)=-\mathrm{a}\mathrm{x}+b}$. (b) Polynomial regression model: ${\displaystyle y\left(x\right)={\mathrm{a}\mathrm{x}}^2-\mathrm{b}\mathrm{x}+c}$. (c) Logistic regression model: ${\displaystyle y\left(x\right)=-\mathrm{a}\mathrm{l}\mathrm{n}\left(x\right)+b}$. (d) Non-linear regression model: ${\displaystyle y\left(x\right)={\mathrm{a}\mathrm{e}}^{-\mathrm{b}\mathrm{x}}}$ that presents the best fit and matches with the proposed analytical solution ${\displaystyle \left[\mathrm{G}\mathrm{l}\mathrm{u}\right]\left(t\right)=e^{-\left(\alpha \right)t}{\left[\mathrm{G}\mathrm{l}\mathrm{u}\right]}_{\mathrm{m}\mathrm{a}\mathrm{x}}}$ . Data: Glucose, Brazil–WB1 (23). Representation of the degree of fit between the numerical solution of the proposed ordinary differential equation (ODE) model and various regression models applied to the same dataset. Panel (a) shows the linear regression, (b) the polynomial regression, (c) the logistic regression, and (d) the non-linear regression model. Notably, the non-linear regression in panel (d) achieves an optimal fit (R² = 1), closely matching the proposed analytical solution. In this panel, the estimated parameter "a"—which represents the initial glucose concentration—is approximately 55.5 mg·g⁻¹ (SE = 3.68e-07), while parameter "b", reflecting the combined rate of glucose consumption by yeasts and lactic acid bacteria, is estimated at about -1.85 (SE = 1.72e-08). The near-zero p-value (<2e-16) and a very small residual standard error (1.33e-06) underscore the statistical significance and precision of these estimates, thereby reinforcing the model's capacity to accurately describe the sugar consumption dynamics during cocoa fermentation.