Statistical protocol for nonlinear model selection using cumulative CO₂ production during alcoholic fermentation by Saccharomyces cerevisiae

Abstract

Understanding the kinetics of alcoholic fermentation is essential for characterizing the metabolic performance of wine yeasts under different environmental conditions. This study aimed to establish a systematic method for selecting nonlinear regression models to describe the cumulative carbon dioxide production by Saccharomyces cerevisiae during alcoholic fermentation at two different temperatures: 18 °C and 28 °C. Fermentations were conducted in synthetic must based on diluted grape must with 250 g L⁻¹ of reducing sugars, inoculated with a commercial yeast strain, and incubated under static conditions. CO₂ production was monitored gravimetrically through daily mass loss. Four sigmoidal models were evaluated: the reparameterized Gompertz model, the classical Gompertz model, the Logistic model, and the No Lag Phase model. Ordinary least squares were used to fit the models, and statistical diagnostics were performed to validate the assumptions of normality, homoscedasticity, and independence of residuals. The reparameterized Gompertz model showed superior statistical and biological performance, yielding significant and coherent estimates of lag phase, maximum CO₂ production rate, and final cumulative CO₂. At 18°C, a distinct adaptation phase was observed, whereas at 28°C, fermentation commenced immediately with higher production rates and total yield. The reparameterized Gompertz model achieved the best fit at both temperatures, with the lowest root mean square error and Bayesian information criterion values, and was the only model to satisfy all residual diagnostic tests. The results demonstrate that temperature exerts a critical influence on yeast kinetics and model adequacy. This study contributes a robust and reproducible protocol for selecting nonlinear models applied to laboratory-scale alcoholic fermentation by S. cerevisiae, supporting improved kinetic characterization under controlled experimental conditions.

Keywords:
CO₂-based kinetic modeling; Predictive kinetic modeling; Nonlinear regression; Statistical model selection; Wine yeast screening; Enological yeast

Highlights

The reparametrized Gompertz model best describes CO2 production at 18 °C and 28 °C

Higher temperatures accelerate fermentation and shorten the lag phase

Statistical tests confirm the robustness and adequacy of the reparametrized Gompertz model

1 Introduction

Alcoholic fermentation is the central biological process in wine production, and a thorough understanding of its dynamics is crucial for producing wines that are both of high quality and typicity. During this process, the yeast S. cerevisiae plays a critical role by converting grape must sugars into ethanol, carbon dioxide (CO₂), and a variety of other metabolites that contribute to the wine's sensory profile. Given its importance, studying the kinetics of alcoholic fermentation is fundamental, and one of the most widely used indicators is the cumulative production of CO₂. This metric offers a straightforward yet effective means of monitoring yeast metabolic activity throughout fermentation (International Organisation of Vine and Wine, 2012).

Despite its practicality, the method outlined in Resolution OIV-OENO 370/2012 by the International Organisation of Vine and Wine (International Organisation of Vine and Wine, 2012) offers only a final snapshot of the fermentation process. It estimates total CO₂ production and converts it into an equivalent quantity of sugar consumed. However, more profound insight into the dynamics of fermentation can be achieved by examining key kinetic parameters, such as the lag phase, during which yeast adapts to its environment; the exponential phase, characterized by peak metabolic activity; and the stationary phase, marked by nutrient depletion. These phases are significantly influenced by factors, including yeast strain, fermentation medium composition, and, especially, temperature (Quirós et al., 2013).

To characterize these kinetic phases more precisely, mathematical models have been increasingly applied. Models such as the Gompertz (in both its classical and reparameterized forms) and the Logistic model allow for the extraction of biologically meaningful parameters, including lag time (λ), the maximum rate of CO₂ production (µmax), and the cumulative maximum output (A) (Zwietering et al., 1990). These models are widely used in predictive microbiology due to their ability to describe the typical sigmoidal shape of fermentation curves. In scenarios where fermentation begins immediately—such as when using well-activated starter cultures or conducting fermentations at elevated temperatures—simpler models like the No Lag Phase model, which omits the adaptation phase, may also be appropriate (Fang et al., 2012).

Nevertheless, selecting the most appropriate nonlinear regression model for biological data is a complex task. It requires more than merely achieving a visually satisfactory curve fit. A robust model selection process must consider the biological interpretability of parameters, the statistical soundness of the fit, and the overall coherence of the model in describing system behavior. Therefore, establishing a logical and systematic approach for selecting nonlinear regression models is particularly important in complex systems, such as alcoholic fermentation driven by wine yeasts.

To address this challenge, Archontoulis & Miguez (2015) proposed a seven-step framework for model selection, guiding the researcher from initial model choice to the final interpretation of estimated parameters. Their methodology emphasizes a rigorous and biologically informed approach that includes careful definition of initial parameter values, assessment of numerical convergence, and residual diagnostics (normality, homoscedasticity, and independence). In addition, the use of performance metrics such as adjusted coefficient of determination (adjusted R²), root mean square error (RMSE), and the Bayesian Information Criterion (BIC) strengthens the selection of the most suitable model.

In this context, the present study aims to establish a systematic procedure for selecting nonlinear regression models applied to the cumulative production of CO₂ during alcoholic fermentation by S. cerevisiae, comparing fermentation kinetics at two distinct temperatures: 18 °C and 28 °C. The thoroughness of our research process ensures the reliability and robustness of our findings.

2 Material and methods

2.1 Initial inoculum preparation and fermentation conditions

The initial inoculum was prepared using the commercial yeast strain Saccharomyces cerevisiae SafOeno™ NDA 21 (Fermentis, France), which was first activated in YEPD medium (yeast extract 10 g L⁻¹, peptone 20 g L⁻¹, glucose 20 g L⁻¹) for 24 h at 28 °C. Subsequently, three successive subcultures were performed in 5 mL of fresh YEPD medium, with 100 µL transfers every 48 h, until the cell concentration reached approximately 10⁸ CFU·mL⁻¹. After growth, 1 mL of the activated culture was inoculated into 100 mL Erlenmeyer flasks containing 50 mL of fermentative solution prepared from concentrated ‘Sangiovese’ grape must diluted to 250 g L⁻¹ of reducing sugars and sterilized by autoclaving. The flasks were sealed with Müller-type gas release valves (airlocks) and incubated under static conditions at 18 °C and 28 °C (Figure 1). In parallel, control flasks containing the same sterilized must (50 mL) but without yeast inoculation were prepared and incubated under identical conditions, also sealed with Müller-type traps. These controls were used to quantify non-fermentative mass losses, such as water evaporation. Fermentation progress was monitored gravimetrically by daily measurements of CO₂ release, calculated from the loss of flask mass until stabilization, following the official OIV gravimetric methodology (OIV-OENO 370/2012). The mass loss observed in the control flasks was subtracted from the total mass loss measured in the fermentation flasks to obtain the corrected CO₂ production values. Results were expressed as cumulative CO₂ mass (g) per 100 mL of must. The experiment was conducted in a completely randomized design with three replicates at each temperature. Variability among replicates was assessed by analysis of variance (ANOVA), and the coefficient of variation (CV%) was calculated from the residual mean square. Mean values and standard deviations were calculated for cumulative CO₂ production at each sampling time. Nonlinear models were fitted separately for each temperature using the mean cumulative CO₂ values obtained from the three replicates.

Figure 1
Schematic representation of the fermentation apparatus equipped with a Müller-type trap for gravimetric CO₂ quantification.

2.2 Nonlinear regression models evaluated

The cumulative CO₂ production curves over time were fitted using nonlinear regression models to describe the fermentation kinetics of S. cerevisiae at different temperatures. The models applied belong to the sigmoidal function family and are characterized by an upper asymptote, typical of substrate-limited processes, and an inflection point corresponding to the maximum fermentation rate. The general structure of the models is defined as (Equation 1 ):

$y_i=F\left(t_i;\theta \right)+{\epsilon }_i$ (1)

Where: $y_i$is the cumulative CO₂ mass (g 100 mL^-1) at time $t_i$ (days); $F\left(t_i;\theta \right)$ is the nonlinear regression function dependent on time and the parameter vector θ; and ${\epsilon }_i$ is the random error associated with observation $i$. It was assumed that ${\epsilon }_i~N\left(\mathrm{0,}{\delta }^2\right),$ with independence, identical distribution, constant variance, and null covariance between observations $i\ne j.$

The estimated parameters theta varied according to the model, generally including the lag phase duration (λ), maximum fermentation rate (µmax), and the maximum cumulative CO₂ production (A). Model fitting was performed using the ordinary least squares method via the R software and the non-linear least square nls() function for iterative parameter estimation (Negrão et al., 2021). The functions fitted to the treatments were:

Reparameterized Gompertz (Zwietering et al., 1990) (Equation 2):

$y=y_0+\left(A-y_0\right)exp\left\{-exp\left[\frac{{\mu }_{max}.e}{A-y_0}\left(\lambda -t\right)+1\right]\right\}+{\epsilon }_i$ (2)

Classical Gompertz (Zwietering et al., 1990) (Equation 3):

$y=Aexp\left\{-exp\left[\frac{{\mu }_{max}.e}{A}\left(\lambda -t\right)+1\right]\right\}+{\epsilon }_i$ (3)

Logistic (Zwietering et al., 1990) (Equation 4):

$y=\frac{A}{\left\{1+exp\left[\frac{4{\mu }_{max}}{A}\left(\lambda -t\right)+2\right]\right\}}+{\epsilon }_i$ (4)

No Lag Phase (Fang et al., 2012) (Equation 5):

$y=y_0+A-\ln \left\{exp\left(y_0\right)+\left[\exp \left(A\right)-\exp \left(y_0\right)\right]\exp \left(-{\mu }_{max}t\right)\right\}+{\epsilon }_i$ (5)

2.3 Initial parameter value definition

To assist in defining initial parameter values for the nonlinear regression models, an interactive application was developed in R using the Shiny, nls(), and nls2 packages. This system enabled iterative fitting of sigmoidal functions to experimental cumulative CO₂ production data over time (Appendix 1). The graphical interface allowed for the manual input of initial values and provided immediate visual feedback of the fitted curve superimposed on the observed data. Additionally, the application delivered a detailed statistical summary of the fitted model, including parameter estimates, standard errors, t-values, statistical significance, residual standard error, iteration count, and convergence tolerance.

2.4 Model fitting and convergence verification

Model fitting was performed using the NonlinearRegression package in R. For each temperature evaluated (18 °C and 28 °C), four sigmoidal growth functions were fitted: reparameterized Gompertz, classical Gompertz, Logistic, and No Lag Phase. Initial parameter values were defined using the interactive application described earlier. Each model was fitted using ordinary least squares via the fit_regression() function with design = 2, allowing for the decomposition of the total sum of squares into regression, lack-of-fit, and residual components. To support model diagnostics and enhance robustness, the nlstools package (Baty et al., 2015) was employed, offering tools for residual analysis, confidence intervals, and graphical assessment of nonlinear regression fits. The statistical significance of each parameter was assessed via F-tests at a 5% significance level.

2.5 Validation of model assumptions

To ensure the statistical validity of the nonlinear models, their underlying assumptions were evaluated through residual analysis. Residuals extracted from the fitted models were subjected to three diagnostic tests: the Shapiro-Wilk test to assess normality, the Breusch-Pagan test to evaluate homoscedasticity, and the Durbin-Watson test to verify the independence of residuals. A model was considered statistically valid only when all three null hypotheses were accepted (p > 0.05), thereby ensuring the reliability of the estimated parameters and the appropriateness of the model for treatment comparisons.

2.6 Model fit quality indicators

Model performance was evaluated using statistical indicators that quantify predictive quality:

Adjusted Coefficient of Determination (R²_adjusted) (Equation 6):

$R_{ajust}^2=1-\left[\frac{\left(1-R^2\right)\left(n-1\right)}{n-k-1}\right]$ (6)

where $n$ is the number of observations and $k$ is the number of model parameters.

Bayesian Information Criterion (BIC) (Equation 7):

$BIC=n.\ln \left(Q{M}_{res}\right)+k.\text{ln}\left(n\right)$ (7)

where $Q{M}_{res}$ is the residual mean square.

Root Mean Square Error (RMSE) (Equation 8):

$RMSE=\sqrt{\frac{1}{n}{\text{Σ}}_{i=1}^n{\left(y_i-y_{i-1}\right)}^2}$ (8)

Models were compared based on these criteria, with the most appropriate being those exhibiting higher R² and R²-Adjusted values, as well as lower BIC and RMSE, reflecting an optimal balance between goodness of fit and model complexity.

3 Results and discussion

Table 1 presents the estimated initial values for the four nonlinear regression models fitted to the cumulative CO₂ production data during alcoholic fermentation by S. cerevisiae at 18°C and 28 °C. These values were obtained using custom R scripts employing the Shiny, nls(), and nls2 packages (Appendix 1). The parameters analyzed included the initial CO₂ production (y₀), maximum cumulative production (A), maximum production rate (µmax), and lag time (λ), all derived directly from model fitting, with statistical significance assessed via Student’s t-test. At 18 °C, the reparameterized Gompertz model yielded significant estimates for y₀ = −0.59 g of CO₂ (p < 0.05), A = 11.38 g of CO₂ 100 mL^-1 (p < 0.001), µmax = 1.74 g of CO₂ day⁻¹ (p < 0.001), and λ= 0.53 days (p < 0.05). The classical Gompertz model estimated λ = 0.97 days (p < 0.001), and the Logistic model estimated λ = 1.191 days (p < 0.001). The No Lag Phase model, which omits the lag phase, produced a non-significant y₀ (p > 0.05), highlighting its biological limitation in capturing fermentations with a pronounced adaptation period.

Table 1
Initial parameter values used to fit nonlinear regression models to cumulative CO₂ production at 18 and 28 °C.

Model	yo	A	µmax	ë
	g of CO2 100 mL-1	g of CO2 100 mL-1	g of CO2 day-1	day
18 °C
Reparameterized Gompertz	-0.59*	11.38***	1.74***	0.53*
Gompertz	-	11.26***	1.77***	0.97***
Logistic	-	10.88***	1.80***	1.19***
No Lag Phase	-0.43ns	10.84***	1.40***	-
28 °C
Reparameterized Gompertz	-1.34*	12.02***	2.95***	0.31ns
Gompertz	-	11.96***	2.93***	0.20*
Logistic	-	11.83***	2.86***	0.29ns
No Lag Phase	-0.33ns	10.78***	2.35***	-

• ns = not significant ( p ≥ 0.05); p < 0.05 (*), p < 0.001 (***) by Student’s t-test applied to the estimated coefficients

At 28 °C, the reparameterized Gompertz model yielded µmax = 2.95 g of CO₂ day⁻¹ (p < 0.001) and A = 12.02 g of CO₂ 100 mL^-1 (p < 0.001), representing increases relative to 18 °C and suggesting enhanced fermentative performance at the higher temperature. However, λ was not statistically significant (p > 0.05), likely reflecting a shortened or absent lag phase at higher temperatures. The logistic and classical Gompertz models yielded similar fits, both with high µmax values and A close to 12 g of CO₂ 100 mL^-1. Logistic. Once again, the No Lag Phase model resulted in non-significant estimates for y₀ and λ, confirming its poor performance in representing the early stages of fermentation kinetics.

The initial values obtained using the interactive R application enabled real-time visualization of parameter effects on the predicted curve, facilitating the selection of biologically plausible starting points based on the morphology of the experimental curve. As discussed by Zhang et al. (2020) and Liu et al. (2008), selecting initial values that are consistent with yeast physiology during fermentation substantially enhances the statistical and physiological robustness of the parameter estimates.

Using the estimated initial values, regression curves were constructed for each nonlinear model using the mean cumulative CO₂ values obtained from the three replicates at each temperature. Fermentation experiments showed low variability among replicates under laboratory conditions. The coefficient of variation (CV), calculated from the residual mean square of the ANOVA, was 2.93% at 18 °C and 0.64% at 28 °C, indicating high experimental reproducibility.

As shown in Figure 2, all models exhibited good overall adherence to the observed data. However, differences became evident in the early stages of fermentation, particularly at 18 °C, where a distinct lag phase was observed. Under these conditions, the reparameterized Gompertz model was the only one that accurately captured the initial delay, aligning well with the experimental data. In contrast, the No Lag Phase model overestimated CO₂ production in the early phase, resulting in noticeable discrepancies.

Figure 2
Nonlinear regression models fitted to cumulative CO₂ production during alcoholic fermentation at 18 °C and 28 °C. The red line represents mean observed values (n = 3), and error bars denote standard deviation. Curves correspond to predictions from the reparameterized Gompertz (RepGom), Gompertz (Gom), Logistic (Log), and no-lag phase (NPL) models.

Combining visual inspection of the curves with the results of the ANOVA presented in Table 2, all models showed significant regression (p < 0.001). However, when considering the regression deviation as a lack-of-fit test, as proposed by Eubank et al. (2005), only the reparameterized Gompertz model at 18 °C showed no significant lack of fit (p > 0.05), indicating that its structural flexibility was sufficient to capture the systematic patterns in the data. The other models exhibited significant deviations (p < 0.05), indicating functional inadequacy.

Table 2
Analysis of variance (ANOVA) and regression deviation for evaluating the fit of nonlinear models to the cumulative CO₂ production curve at 18 °C and 28 °C.

Temperature	Model	GL	SQ	QM	FC
18 °C	Reparameterized Gompertz
	Regression	3	647.116	215.705	4972.085*
	Regression Deviation	10	0.56118	0.05612	0.2812ns
	Gompertz
	Regression	2	646.570	323.285	7451.827*
	Regression Deviation	11	1.10793	0.10072	2.3216*
	Logistic
	Regression	2	641.734	320.867	7396.092*
	Regression Deviation	11	5.94387	0.54035	12.4552*
	No Lag Phase
	Regression	2	640.160	320.080	7377.953*
	Regression Deviation	11	7.51765	0.68342	15.7531*
28 °C	Reparameterized Gompertz
	Regression	3	601.655	200.551	56342.312*
	Regression Deviation	10	0.77553	0.07755	21.7875*
	Gompertz
	Regression	2	600.753	300.376	84386.762*
	Regression Deviation	11	1.67756	0.15251	42.8444*
	Logistic
	Regression	2	596.934	298.467	83850.323*
	Regression Deviation	11	5.49650	0.49968	140.3789*
	No Lag Phase
	Regression	2	597.632	298.816	83948.405*
	Regression Deviation	11	4.79825	0.4362	122.5457*

• ns = not significant (p ≥ 0.05); * = significant (p < 0.05) by the F-test.

Table 3 presents the results of diagnostic tests applied to the residuals of the fitted nonlinear models to evaluate the assumptions of normality (Shapiro-Wilk), homoscedasticity (Breusch-Pagan), and independence (Durbin-Watson). Validating these assumptions is critical for ensuring the reliability of the parameter estimates and the validity of statistical inference. At 18°C, the reparameterized Gompertz model was the only one to meet all assumptions, with p-values greater than 0.05 in all tests. The classical Gompertz model violated the homoscedasticity assumption (p = 0.0446), suggesting non-constant residual variance. The Logistic model violated both homoscedasticity and independence (p = 0.0410 and p = 0.0020, respectively), compromising the reliability of its estimates. The No Lag Phase model marginally failed the independence test (p = 0.038), though it satisfied the other assumptions.

Table 3
Results of normality, homoscedasticity, and autocorrelation tests on the residuals of the models evaluated at 18 °C and 28 °C.

Model	18 °C			28 °C
	Shapiro-Wilk (p)	Breusch-Pagan (p)	Durbin-Watson (p)	Shapiro-Wilk (p)	Breusch-Pagan (p)	Durbin-Watson (p)
Reparameterized Gompertz	0.1804 ns	0.0537 ns	0.8180 ns	0.1115 ns	0.9116 ns	0.9780 ns
Gompertz	0.3200 ns	0.0446 *	0.0820 ns	0.3752 ns	0.0019 *	0.8620 ns
Logistic	0.5451 ns	0.0410 *	0.0020 *	0.2773 ns	0.0013 *	0.2440 ns
No Lag Phase	0.4798 ns	0.1331 ns	0.0380 *	0.2347 ns	0.2332 ns	0.0400 *

• ^ns
  = not significant (p ≥ 0.05); * = significant (p < 0.05). Tests: Shapiro–Wilk (normality), Breusch–Pagan (homoscedasticity), and Durbin–Watson (independence).

At 28 °C, the reparameterized Gompertz model again met all criteria. Conversely, both the classical Gompertz and logistic models exhibited significant heteroscedasticity (p < 0.01), with the Logistic model also showing signs of autocorrelation (though not substantial, p = 0.244). The No Lag Phase model violated the independence assumption (p = 0.04), despite yielding acceptable results in the other tests. These findings highlight the reparameterized Gompertz model’s superior performance in terms of residual structure adequacy across both temperatures. Autocorrelation or heteroscedasticity in the different models may compromise the accuracy of kinetic parameter estimates, such as µmaxand λ, thus justifying the use of stringent statistical criteria for model selection.

According to the results in Table 4, the reparameterized Gompertz model produced statistically significant and biologically consistent estimates for all kinetic parameters at both temperatures. At 18 °C, it had the lowest RMSE (0.116) and BIC (−7.490), indicating an excellent fit to the experimental data. At 28 °C, the estimated maximum CO₂ production rate (µmax = 2.95 g of CO₂ day⁻¹) confirmed the positive effect of temperature on the fermentative activity of S. cerevisiae. In comparison, the classical Gompertz model, though biologically plausible, showed higher residual dispersion and less parsimony, as indicated by its higher RMSE and BIC values.

Table 4
Estimated kinetic parameters and goodness-of-fit indicators for nonlinear regression models of cumulative CO₂ production by S. cerevisiae at 18 °C and 28 °C.

Model	yo	A	µmax	ë	R2aj		BIC	RMSE
	g of CO2 100 mL-1	g of CO2 100 mL-1	g of CO2 day-1	day
18 °C
Reparameterized Gompertz	-0.59**	11.38***	1.74***	0.53**		0.999	-7.490	0.116
Gompertz	-	11.26***	1.77***	0.97***		0.998	-0.606	0.163
Logistic	-	10.88***	1.80***	1.19***		0.988	22.912	0.377
No Lag Phase	-0.43***	10.84***	1.40***	-		0.989	19.586	0.344
28 °C
Reparameterized Gompertz	-1.34***	12.02***	2.95***	0.31***		0.998	-2.960	0.136
Gompertz	-	11.96***	2.93***	0.20***		0.996	-0.606	0.200
Logistic	-	11.83***	2.86***	0.29***		0.988	21.816	0.361
No Lag Phase	-0.33ns	10.78***	2.35***	-		0.993	10.861	0.244

• ^ns
  = not significant (p >= 0.05); ** = p < 0.01; *** = p < 0.001 by Student’s t-test applied to the estimated coefficients.

The Logistic and No Lag Phase models performed poorly, particularly at 18 °C. The Logistic model had the highest RMSE (0.377), reflecting low predictive accuracy. While the No Lag Phase model improved slightly at 28 °C, likely due to the diminished lag phase under optimized conditions, it still fell short of the reparameterized Gompertz model. Therefore, from both statistical and biological perspectives, the reparameterized Gompertz model emerges as the most appropriate for modeling alcoholic fermentation kinetics under the experimental conditions. These findings also corroborate previous studies, which show that the maximum specific growth rate (µmax) tends to increase with temperature up to an optimal threshold (Cruz et al., 2012; Coleman et al., 2007), highlighting the regulatory role of temperature in fermentation kinetics.

By applying the reparameterized Gompertz model throughout the fermentation process, we obtained the cumulative CO₂ production curves for temperatures of 18 °C and 28 °C (Figure 3). In both cases, the model effectively captured the fermentation dynamics, characterized by a rapid rise in CO₂ production followed by a plateau. However, the curve at 28 °C was shifted upward compared to 18 °C, with maximum CO₂ production increasing from 11.38 to 12.02 g of CO₂ 100 mL^-1 (Table 1), and µmax rising from 1.74 to 2.95 g of CO₂ day⁻¹, indicating a faster fermentation process at the elevated temperature. The λ also decreased from 0.53 to 0.31 days, reflecting faster yeast adaptation. These results indicate that higher temperatures favor both the onset and intensity of alcoholic fermentation, consistent with the temperature-sensitive mechanisms regulating the metabolism of S. cerevisiae in wine musts.

Figure 3
Fit of the reparameterized Gompertz model to the cumulative CO₂ production curve during alcoholic fermentation of ‘Sangiovese’ grape must by S. cerevisiae at 18 °C and 28 °C.

Thus, fermentation temperature is a key determinant of the kinetic behavior of S. cerevisiae, directly influencing the statistical significance and biological interpretability of parameters estimated through nonlinear regression models, such as the reparameterized Gompertz, classical Gompertz, Logistic, and No Lag Phase models. As evidenced in this study, increasing the temperature from 18 °C to 28 °C significantly enhanced µmaxand A, while reducing λ. This pattern aligns with prior findings (Li & Deed, 2021), which show that higher temperatures shorten lag duration by accelerating yeast adaptation and transition to exponential growth. The lack of statistical significance for λ at 28°C may reflect this natural physiological response, driven by increased enzymatic activity and metabolic rate in response to thermal stimulation (Brandão et al., 2020; Ganucci et al., 2018).

Beyond biological effects, temperature also impacts the statistical robustness of model fits. Parameters such as y₀ and λ may lose significance under accelerated fermentation conditions, as observed at 28 °C, where the No Lag Phase model performed poorly in capturing the adaptation phase. Model fit quality, as indicated by metrics such as adjusted R² and BIC, was highest for the reparameterized Gompertz model, particularly under conditions where the CO₂ production curve maintained a well-defined sigmoidal shape. These findings underscore the importance of accounting for temperature as a key modulatory factor in both model selection and parameter interpretation. Furthermore, they highlight the need to integrate environmental variables into kinetic modeling frameworks to adequately represent fermentation dynamics under controlled experimental conditions. Although mathematical models have been successfully applied to broader predictive contexts (Silva et al., 2024), extrapolation of the present results to pilot or industrial-scale fermentations requires additional validation, given the increased operational complexity of large-scale systems.

Therefore, when accounting for thermal effects on parameter estimation, it becomes evident that models capable of capturing yeast physiological responses under varying environmental conditions are essential. As noted by Liu et al. (2014), the Gompertz model shows superior predictive performance within temperature ranges compatible with yeast physiology. In contrast, extrapolations to extreme temperatures may require alternative models or more structurally robust approaches.

4 Conclusions

This study proposed a systematic protocol for selecting nonlinear regression models to describe CO₂ production kinetics during alcoholic fermentation by S. cerevisiae. Among the models tested, the reparameterized Gompertz model showed the best statistical and biological performance, accurately estimating lag phase (λ), maximum CO₂ production rate (µmax), and cumulative yield (A) at both 18 °C and 28 °C. The protocol combined residual diagnostics and fit quality indicators (BIC, RMSE) to ensure model adequacy. The results highlight temperature as a key factor influencing fermentation kinetics and reinforce the suitability of the reparameterized Gompertz model for modeling yeast metabolism under enological conditions, providing a robust tool for strain characterization and process optimization.

Appendix 1

library(shiny); library(nls2)

dados <- data.frame(Tempo = 0:13, CO2 = c(0, 0.353, 2.093, 3.763, 5.48, 6.943, 8.18, 9.003, 9.78, 10.213, 10.58, 10.813, 11.057, 11.317))

modelo_gompertz <- function(a, b, c, d) {

nls(CO2 ~ a + (c - a) * exp(-exp(((d * exp(1)) / (c - a)) * (b - Tempo) + 1)),

data = dados, start = list(a = a, b = b, c = c, d = d))

}

ui <- fluidPage(

titlePanel("Ajuste Gompertz"),

sidebarLayout(

sidebarPanel(

numericInput("a", "a", 0), numericInput("b", "b", 5),

numericInput("c", "c", 12), numericInput("d", "d", 2),

actionButton("ajustar", "Ajustar")

),

mainPanel(plotOutput("plot"), verbatimTextOutput("summary"))

)

)

server <- function(input, output) {

modelo <- eventReactive(input$ajustar, {

tryCatch(modelo_gompertz(input$a, input$b, input$c, input$d), error = function(e) NULL)

})

output$plot <- renderPlot({

plot(dados, col = "blue", pch = 16, main = "Ajuste Gompertz", xlab = "Tempo", ylab = "CO2")

if (!is.null(modelo())) lines(dados$Tempo, predict(modelo()), col = "red", lwd = 2)

})

output$summary <- renderPrint({ if (!is.null(modelo())) summary(modelo()) else "Erro no ajuste" })

}

shinyApp(ui = ui, server = server)

Acknowledgements

The authors gratefully acknowledge the financial support of the Instituto Federal de Santa Catarina (IFSC) for funding the project, and the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for granting a PIBITI research scholarship.

Data Availability

All data generated or analyzed in this study are included in this published article.

• Cite as:
  Sousa, G. A., Raulino, G., Jesus, J., Mathias, J. G., & Stroschein, M. R. D. (2026). Statistical protocol for nonlinear model selection using cumulative CO₂ production during alcoholic fermentation by Saccharomyces cerevisiae. Brazilian Journal of Food Technology, 29, e2025156. https://doi.org/10.1590/1981-6723.2025156
• Funding:
  This work was supported by the Instituto Federal de Santa Catarina (IFSC) through internal project funding (project number PIURP3843 /2024).

References

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  » https://doi.org/10.2134/agronj2012.0506
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Edited by

• Section Editor:
  Mateus Petrarca.

Publication Dates

• Publication in this collection
  24 July 2026
• Date of issue
  2026

History

• Received
  11 Dec 2025
• Accepted
  23 Apr 2026